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+CMLCompiler: A Unified Compiler for Classical Machine
+Learning
+Xu Wen
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+wenxu@ict.ac.cn
+Wanling Gao
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+gaowanling@ict.ac.cn
+Anzheng Li
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+lianzheng20g@ict.ac.cn
+Lei Wang
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+wanglei_2011@ict.ac.cn
+Zihan Jiang
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+jiangzihan@ict.ac.cn
+Jianfeng Zhan∗
+Institute of Computing Technology,
+Chinese Academy of Sciences
+University of Chinese Academy of
+Sciences
+zhanjianfeng@ict.ac.cn
+ABSTRACT
+Classical machine learning (CML) occupies nearly half of machine
+learning pipelines in production applications. Unfortunately, it fails
+to utilize the state-of-the-practice devices fully and performs poorly.
+Without a unified framework, the hybrid deployments of deep learn-
+ing (DL) and CML also suffer from severe performance and porta-
+bility issues. This paper presents the design of a unified compiler,
+called CMLCompiler, for CML inference. We propose two unified
+abstractions: operator representations and extended computational
+graphs. The CMLCompiler framework performs the conversion and
+graph optimization based on two unified abstractions, then outputs
+an optimized computational graph to DL compilers or frameworks.
+We implement CMLCompiler on TVM. The evaluation shows CML-
+Compiler’s portability and superior performance. It achieves up to
+4.38× speedup on CPU, 3.31× speedup on GPU, and 5.09× speedup
+on IoT devices, compared to the state-of-the-art solutions — scikit-
+learn, intel sklearn, and hummingbird. Our performance of CML
+and DL mixed pipelines achieves up to 3.04x speedup compared
+with cross-framework implementations.
+CCS CONCEPTS
+• Computing methodologies → Machine learning; • Computer
+systems organization → Real-time systems.
+KEYWORDS
+Classical Machine Learning, Deep Learning, Compiler
+1
+INTRODUCTION
+Deep learning (DL) and classical machine learning (CML), collec-
+tively called machine learning (ML), have played an increasingly
+critical role in recent years. DL refers to those neural network mod-
+els, such as convolutional neural networks (CNNs) [24], recurrent
+neural networks (RNNs) [28], and generative adversarial networks
+(GANs) [16]. Different from DL, CML represents a set of non-neural
+network models in ML, e.g., linear models [37], decision trees [26],
+∗Corresponding author.
+Hardware
+CPU
+GPU
+IoT
+...
+Models
+Linear Models
+Trees
+SVMs
+...
+Compiler Framework
+Unified Abstractions
+CMLCompiler
+DL Frameworks (PyTorch)
+PyTorch Runtime
+DL compilers (TVM)
+TVM Runtime
+DL Frameworks (PyTorch)
+PyTorch Runtime
+DL compilers (TVM)
+TVM Runtime
+Figure 1: The CMLCompiler design. Our contributions are
+highlighted in green color.
+random forests [4], and support vector machines [42]. DL stands
+out because of its accuracy, while CML is still widely used for lower
+time and energy costs. Doris Xin et al. [47] analyze 3000 produc-
+tion ML pipelines at Google and find that 40% of them use CML
+models. Besides, many real-world applications adopt hybrid de-
+ployments of CML and DL [2] to guarantee high accuracy and low
+latency [25, 27, 36, 38], e.g., DL models for feature embedding and
+CML models for classification or regression.
+DL compilers, like TVM [7, 10, 23], provide a structural approach
+to tackle the portability issue and facilitates wide deployment of DL
+models on a broad spectrum of devices like GPUs, FPGAs, and IoT
+devices and guarantees an appreciable performance. DL compilers
+use computational graphs as high-level abstractions, supporting
+a large variety of DL models. Meanwhile, DL compilers propose
+low-level abstractions such as tensor representation to generate
+executable code. For newborn hardware, the vendor just need to
+provide hardware primitives, instead of a sophisticated high per-
+formance library that is prohibitively costly. Based on the tensor
+1
+arXiv:2301.13441v1  [cs.LG]  31 Jan 2023
+
+Xu Wen et al.
+representation and computational graphs abstractions, many opti-
+mizations [8, 22, 49] are proposed to boost performance, e.g., they
+provide sophisticated support for CPU processor architectures as
+the latter has different architectures, diverse core numbers, ex-
+tended instructions, and cache sizes.
+However, despite its popularity and importance, CML suffers
+from severe portability and performance issues. State-of-the-practice
+and state-of-the-art CML frameworks [17, 29, 32] provide ad-hoc
+solutions, implementing each CML model on every hardware device
+case by case due to the lack of unified abstractions. These ad-hoc
+solutions raise considerable difficulties in developing a general-
+purpose framework and optimization techniques to achieve optimal
+performance for every model. They either lack the support or only
+partially support various hardware devices, such as GPUs, FPGAs,
+and IoT devices. In addition, adding support for a model on a new
+hardware device needs great effort, more than several thousands
+of lines of codes [13], let alone hundreds or thousands of models
+and devices. Moreover, they also face performance issues. Even on
+the CPUs – the most popular CML platform, the performance is
+unsatisfactory due to the lack of specific optimizations for advanced
+characteristics like multi-cores and SIMD. The hybrid deployment
+of CML and DL models faces more severe problems.
+Our intuition is to enable CML to leverage DL’s well-defined
+unified abstractions and highly mature compilers, optimization
+technologies, and frameworks. Unfortunately, it is not a trivial task.
+There are significant distinctions in operators and models between
+CML and DL. DL operators focus on tensors, while CML handles ar-
+rays, matrices, scalars, and tables. DL models are all neural network
+models, while CML models, such as decision trees and SVMs, can
+hardly be represented as neural networks. Most DL models are ex-
+pressible as flat sequences of operations without if-statements [35],
+but if-statements frequently occur in CML models. Existing DL ab-
+stractions, such as tensor representation and computational graphs,
+can not directly represent CML operators and models. Those dis-
+tinctions determine CML can hardly leverage the DL ecosystems
+directly. Several efforts attempt to support CML models on DL
+frameworks, e.g., TensorFlow [1] provides a CPU-based decision
+forest library TF-DF [43]. However, these attempts do not solve
+the generality and portability issue. They only support a narrower
+range of models, lacking support for GPUs and IoT devices.
+This paper focuses on CML inference for the first step, consid-
+ering its great significance that occupies nearly half of the total
+cost [2] and its wide applications in online serving, Internet of
+things (IoT), etc [18, 46]. We will extend our work to CML training
+in the near future. As illustrated in Fig. 1, we propose a unified
+compiler, CMLCompiler, for CML inference, which enables CML to
+leverage the mature DL ecosystems. At the core of CMLCompiler
+are two unified abstractions: operator representations and extended
+computational graphs (ECGs) and a compiler framework. Operator
+representations convert CML operators into tensor formats, while
+an ECG organizes these converted operators in an optimization-
+friendly way. The two unified abstractions define how to convert
+and translate CML models into DL computational graphs, which
+can be recognized and executed by DL frameworks and compilers.
+The CMLCompiler framework consists of four modules – opera-
+tor converter, model parser, graph optimizer, and graph translator.
+The CMLCompiler framework performs the conversion and graph
+optimization based on two unified abstractions, then outputs an
+optimized DL computational graph to DL compilers or frameworks.
+CMLCompiler can also optimize the mixed pipelines of CML and DL.
+As TVM provides portability and sophisticated optimizations, we
+choose to implement CMLCompiler on TVM. Currently, it supports
+up to 35 CML models.
+This paper makes the following contributions:
+• We propose two unified abstractions – operator represen-
+tations and extended computational graphs– to represent
+CML operators and models.
+• We present the design of CMLCompiler, a unified compiler
+for CML inference, based on these abstractions. The CML-
+Compiler framework performs the conversion and graph
+optimization based on two unified abstractions, then outputs
+an optimized DL computational graph to DL compilers or
+frameworks.
+• CMLCompiler enables the hybrid deployment of CML and
+DL with a unified framework.
+• We implement CMLCompiler on top of TVM, achieving up
+to 4.38x speedup on CPU, 3.31x speedup on GPU, and 5.09x
+speedup on IoT devices, compared to the state-of-the-art
+solutions — scikit-learn, intel sklearn, and hummingbird. Our
+support for CML and DL mixed pipelines achieves up to 3.04x
+speedup compared with cross-framework implementations.
+The remainder of the paper is organized as follows. Section 2
+introduces the motivation. Section 3 introduces unified abstractions.
+Section 4 shows design and implementation. Section 5 presents our
+evaluation. Section 6 illustrates the related work. Finally, we draw
+a conclusion in Section 7.
+2
+MOTIVATION
+CML faces severe portability and performance issues. Fig. 2 com-
+pares the performance of sklearn, the most widely used CML frame-
+work on GitHub [33]— against CMLCompiler leveraging DL com-
+pilers. We find that sklearn can not support GPUs and only supports
+IoT devices partially. Adding support for a new hardware device
+needs great effort due to the ad-hoc implementations. For exam-
+ple, adding support for random forest on GPU needs 2.7k lines
+of code [13]. Many models and hardware devices need to be sup-
+ported, requiring hundreds or thousands of more effort. Moreover,
+due to the lack of compilation support for CPU’s features, sklearn
+has poor performance. As shown in Fig .2, CMLCompiler achieves
+2.3x speedup by utilizing AVX2 through compilation compared
+with sklearn. Other CML frameworks such as Spark MLlib [29] and
+H2O [17] face the same problems. Our solution is to propose uni-
+fied abstractions to utilize DL compilers and frameworks, achieving
+portability and high performance.
+CML and DL models are often deployed hybrid in NLP [36], in-
+telligent healthcare [38], recommendation systems [25], etc., espe-
+cially in the scenarios with limited computational power and small
+datasets. Many of them are deployed on heterogeneous hardware
+devices for online serving. As there is no unified system, different
+frameworks are deployed with three disadvantages. First, this lim-
+its the portability. If one framework fails on the target device, the
+whole pipeline corrupts. Second, there are extra costs due to data
+2
+
+CMLCompiler: A Unified Compiler for Classical Machine Learning
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+Figure 2: This figure compares the performance of sklearn,
+the most widely used CML framework on GitHub [33]—
+against CMLCompiler. Our evaluation shows that sklearn
+suffers from both performance and portability issues for a
+lack of unified abstractions.
+conversions across frameworks. Third, it is hard to make optimiza-
+tions across different frameworks. Using a unified framework can
+overcome these disadvantages, so we add the support for hybrid
+deployment of CML and DL in CMLCompiler.
+3
+THE UNIFIED ABSTRACTIONS
+CMLCompiler takes CML models as input and returns DL compu-
+tational graphs as output, utilizing DL frameworks or compilers
+to compile and deploy them. At the core of CMLCompiler are two
+unified abstractions. Operator representations are used to represent
+CML operators in tensor format, as shown in Section 3.1. Extend
+computational graph (ECG) organizes operator representations in
+an optimization-friendly way and can be used to represent CML
+models, as shown in Section 3.2. Section 3.3 shows the supported
+algorithms and extensions for other algorithms.
+3.1
+Operator Representation
+An operator representation uses a combination of one or more DL
+operators with tensors as input and output to represent a CML oper-
+ator. We convert CML operators into DL operators and wrap them
+in the format of operator representations. Data in CML has mainly
+four formats: arrays, matrices, scalars, and tables [44]. Matrices
+and arrays are regarded as two types of tensors whose operators
+can naturally be converted into DL operators. When CML models
+deal with tables, they take numeric data from tables and operate
+it, which can also be regarded as scalars. Hereby, we focus on the
+operators on scalars.
+3.1.1
+Operator categories and corresponding representations. As
+shown in Table 1, we classify CML operators into six categories
+and provide operator representations, respectively.
+(1) Assignment operators assign values to variables. If we assign
+n values 𝑣1, 𝑣2, ..., 𝑣𝑛 to n variables 𝑥1, 𝑥2, ..., 𝑥𝑛, we organize
+these variables and values in two tensors 𝑋 = [𝑥1,𝑥2, ...,𝑥𝑛] and
+𝑉 = [𝑣1, 𝑣2, ..., 𝑣𝑛]. Then we assign tensor V to tensor X to replace
+n scalar assignments. Tensor assignments benefit memory copy
+which stores data in block.
+(2) Swap operators swap two or more variables. These variables
+can be represented in a tensor format and use reorganization oper-
+ators such as 𝑟𝑒𝑠ℎ𝑎𝑝𝑒 to swap the elements.
+(3) Basic arithmetic operators refers to those arithmetic calcu-
+lations based on scalars, such as 𝑎𝑑𝑑, 𝑠𝑢𝑏, 𝑚𝑢𝑙 and 𝑑𝑖𝑣. We use
+element-wise arithmetic operators based on tensors to replace them,
+which can utilize SIMD instructions better.
+(4) Aggregation operators refer to operators that calculate ag-
+gregates among many scalars, such as 𝑚𝑖𝑛, 𝑚𝑎𝑥, 𝑠𝑢𝑚, and 𝑎𝑣𝑔.
+Reduction operators can be used to accomplish that.
+(5) Comparison operators make a comparison between scalars
+and return True or False, such as 𝑙𝑒𝑠𝑠, 𝑒𝑞𝑢𝑎𝑙, and 𝑔𝑟𝑒𝑎𝑡𝑒𝑟. Compar-
+isons with the same operator can be represented in a tensor format
+and use an element-wise comparison to replace.
+(6) Conditional operators are used to represent if-else statements,
+in the form of𝑖𝑓 (𝑒𝑥𝑝𝑟1) 𝑒𝑥𝑝𝑟2𝑒𝑙𝑠𝑒 𝑒𝑥𝑝𝑟3, where𝑒𝑥𝑝𝑟1 is a compar-
+ison operator. If 𝑒𝑥𝑝𝑟2 and 𝑒𝑥𝑝𝑟3 are all assignment or arithmetic
+operators, we convert all three expressions into tensors. However,
+the situation gets tricky if one of 𝑒𝑥𝑝𝑟2 or 𝑒𝑥𝑝𝑟3 is still a conditional
+operator. We call those operators sequential conditional operators.
+Sequential conditional operators may contain many conditions,
+where each element in a tensor may have quite different decision
+paths. The complexity of decision paths makes it difficult to con-
+vert those operators into tensor operators. Those frequent if-else
+statements perform poorly on hardware devices such as GPUs and
+ASICs. Sequential conditional operators are the most delicate, and
+we defer their discussion later.
+3.1.2
+Conditional operators representation. We analyze those widely
+used CML models and find that sequential conditional operators
+mainly occur in tree-based models. So we use decision tree as an
+example to introduce the representation of conditional operators in
+detail, as shown in Fig. 3. We use the combination of DL operators
+to represent those sequential conditional operators.
+The left is a decision tree. The input data is a list of samples;
+each has many features. 𝐼 refers to internal nodes, numbered in the
+order of Level Order Traversal. Each internal node is a conditional
+operator, making a comparison between a feature 𝐹𝑗 and a constant
+threshold 𝑇𝑖. 𝐿 refers to leaf nodes, numbered in the order of In-
+Order Traversal. Each leaf node is an assignment operator, reaching
+which node determines the final result.
+The right in Fig. 3 shows the operator representation, whose
+definitions and properties of weights are shown in Table 2. Input
+data multiplied by 𝑊1 returns those features used in internal nodes
+in an appropriate order. Comparing with 𝑊2 returns the choice of
+each internal node: 0 means left and 1 means right. These choices
+are multiplied by 𝑊3 and then use 𝑎𝑟𝑔𝑚𝑎𝑥 to return the first index
+of the maximum values for each row. For each sample 𝑥𝑘, that index
+is the leaf node 𝑥𝑘 reaches, as proved in appendix A.
+3.1.3
+The features of CML operator representations. As described
+above, we represent CML operators in the format of operator rep-
+resentations. These operator representations have unique features
+different from operators in DL models.
+First, the weights of DL operators and CML operator represen-
+tations have different meanings. The weights in DL models are all
+learnable parameters. Without approximate optimizations such as
+pruning and quantization, those weights are dense, and the data
+type (dtype) should be float32 to ensure accuracy. Many weights of
+CML operator representations have other meanings, such as repre-
+senting the structure of conditional operators. Those weights are
+sparse and can naturally be expressed as low-precision dtypes such
+3
+
+Xu Wen et al.
+Table 1: The summary of operator representation. Each operator representation represents a CML operator. Scalars are marked
+as lower-case letters, while tensors are marked as upper-case letters. EW is short for element-wise.
+CML operators in scalar format
+Operator Representation in tensor format
+Operator Type
+Expressions
+Operator Type
+Expressions
+Assignment
+𝑥1 ← 𝑣1; 𝑥2 ← 𝑣2; ...; 𝑥𝑛 ← 𝑣𝑛
+Assignment
+𝑋 = [𝑥1,𝑥2, ...,𝑥𝑛]; 𝑉 = [𝑣1, 𝑣2, ..., 𝑣𝑛]; 𝑋 ← 𝑉
+Swap
+𝑥1 ← 𝑥2; 𝑥2 ← 𝑥1;
+Reorganization
+𝑋 = [𝑥1,𝑥2]; 𝑟𝑒𝑠ℎ𝑎𝑝𝑒(𝑋);
+Basic Arithmetic
+𝑥1 + 𝑦1; 𝑥2 + 𝑦2; ...; 𝑥𝑛 + 𝑦𝑛
+EW Arithmetic
+𝑋 = [𝑥1,𝑥2, ...,𝑥𝑛]; 𝑌 = [𝑦1,𝑦2, ...,𝑦𝑛]; 𝑋 + 𝑌
+Aggregation
+𝑠𝑢𝑚(𝑥1,𝑥2, ...,𝑥𝑛)
+Reduction
+𝑋 = [𝑥1,𝑥2, ...,𝑥𝑛]; 𝑠𝑢𝑚(𝑋)
+Comparison
+𝑥1 < 𝑦1; 𝑥2 < 𝑦2; ...; 𝑥𝑛 < 𝑦𝑛
+EW Comparsion
+𝑋 = [𝑥1,𝑥2, ...,𝑥𝑛]; 𝑌 = [𝑦1,𝑦2, ...,𝑦𝑛]; 𝑋 < 𝑌
+Conditional
+𝑖𝑓 (𝑒𝑥𝑝𝑟1) 𝑒𝑥𝑝𝑟2 𝑒𝑙𝑠𝑒 𝑒𝑥𝑝𝑟3
+Described in Section 3.1.2
+F5 < T1
+F1  < T2
+F4 < T3
+L2
+F2 <  T4
+L1
+L3
+L4
+L5
+True
+False
+I1
+I2
+I3
+I4
+F5 < T1
+F1  < T2
+F4 < T3
+L2
+F2 <  T4
+L1
+L3
+L4
+L5
+True
+False
+I1
+I2
+I3
+I4
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+Input
+Output
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+Input
+Output
+Figure 3: An example of conditional operator representation in decision tree, a typical classical machine learning model. 𝐹, 𝑇,
+𝐼, and 𝐿 refer to features, thresholds, internal nodes, and leaf nodes. 𝑊1, 𝑊2, and 𝑊3 are the weights of DL operators, whose
+definitions and properties are shown in Table 2, matmul is short for matrix multiplication.
+Table 2: The properties of weights in Fig. 3. 𝑁𝑆, 𝑁𝐹 , 𝑁𝐼 , and
+𝑁𝐿 refer to the number of samples, features, internal nodes,
+and leaf nodes, respectively. 𝐼𝑛𝑝𝑢𝑡 ∈ R𝑁𝑆×𝑁𝐹 means 𝑁𝑆 sam-
+ples, each has 𝑁𝐹 features. 𝑊1 ∈ {0, 1}𝑁𝐹 ×𝑁𝐼 captures the re-
+lationship between features and internal nodes. 𝑊2 ∈ R𝑁𝐼
+is the thresholds used in internal nodes. 𝑊3 ∈ {0, 1}𝑁𝐼 ×𝑁𝐿
+represents the structure between internal nodes and leaf
+nodes. 𝑂𝑢𝑡𝑝𝑢𝑡 ∈ N𝑁𝑆 returns the leaf node index each sam-
+ple reaches. Dtype is the data type of weights. Sparsity is the
+ratio of non-zero data to all data in weights.
+Definition
+Dtype
+Sparsity
+𝑊1[𝑖][𝑗] =
+� 1, 𝐹𝑖 ∈ 𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛(𝐼𝑗)
+0, otherwise
+bool
+1
+𝑁𝐹
+𝑊2[𝑖] = 𝑇ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑(𝐼𝑖)
+float32
+1
+𝑊3[𝑖][𝑗] =
+� 0, 𝐿𝑗 ∈ 𝐿𝑒𝑓 𝑡𝑆𝑢𝑏𝑇𝑟𝑒𝑒(𝐼𝑖)
+1, otherwise
+bool
+[ 1
+2, 1 −
+1
+𝑁𝐿 ]
+as bool. The natural sparse features bring optimizations described
+in Section 4.3.2.
+Second, the frequent operators in DL and CML are not the same.
+Almost all operators in DL take float32 as input and return float32
+as output. CML uses many comparison operators, such as 𝑙𝑒𝑠𝑠,
+𝑒𝑞𝑢𝑎𝑙, and 𝑔𝑟𝑒𝑎𝑡𝑒𝑟, which rarely occur in DL models. Those com-
+parison operators take float or integer as input and return bool
+tensors, bringing remarkable changes in the dtype of input and
+output, which can be used to make optimizations as described in
+Section 4.3.1. Both DL and CML models use indices operators, which
+compare input and returns indices, such as 𝑎𝑟𝑔𝑠𝑜𝑟𝑡 and 𝑎𝑟𝑔𝑚𝑎𝑥.
+Those indices operators have mathematical properties that can
+be used to make graph-level optimizations, as described in Sec-
+tion 4.3.3. These optimizations can be ignored in DL models with
+dozens or hundreds of layers but are helpful for those CML models
+with fewer layers.
+3.2
+Extended Computational Graph
+This section introduces extended computational graph (ECG), which
+organizes operator representations in an optimization-friendly way
+and can be used to represent CML models. ECG is an extension
+based on DL computational graph. In general, a DL computational
+graph is represented as a directed graph where nodes represent
+operations on tensors or program inputs and edges represent data
+dependencies between operations [7]. From a perspective of the DL
+frameworks and compilers, computational graphs are dense and
+float32 by default, such as neural network models. Using approxi-
+mate optimizations like pruning and quantization brings sparse and
+low-precision data to all operators and weights. These optimiza-
+tions cause a decrease in accuracy and bring extra computation,
+such as calibration. When we convert CML operators to operator
+representations, part of those converted operators and weights are
+sparse and low-precision naturally. Using DL computational graphs
+to represent CML models directly is not precise enough and ignores
+many optimization opportunities due to the data type and sparse
+features. So we extend the computational graph in the DL systems
+into extended computational graph (ECG) as the unified abstraction
+for CML models.
+Before introducing ECG, first, we present more details about
+data type (dtype) and sparsity. We define the partial order relation
+for dtypes used in our work:
+𝑓 𝑙𝑜𝑎𝑡32 > 𝑖𝑛𝑡32/𝑓 𝑙𝑜𝑎𝑡16 > 𝑖𝑛𝑡16 > 𝑖𝑛𝑡8 > 𝑖𝑛𝑡4 > 𝑏𝑜𝑜𝑙
+4
+
+CMLCompiler: A Unified Compiler for Classical Machine Learning
+Table 3: Operators used in ECGs
+Operator Type
+Examples
+Comparison
+less, equal, greater, less_equal
+Indices
+argmax, argmin, argsort, argwhere
+Monotonic
+sigmoid, softmax, relu, tanh, exp
+Reduction
+sum, max, min, avg, all, any
+Arithmetic
+gemm, conv, pool
+The lower dtype can be converted into a higher dtype without
+accuracy loss, while a backward conversion with accuracy loss is
+forbidden. Using lower dtype computation, such as int8 matmul,
+can speed up and reduce memory usage. However, there are many
+limitations to dtype optimization. For example, the inputs of the
+same operator should have the same dtype; thus, the dtype of opera-
+tors depends on the largest dtype of inputs. Besides, many hardware
+devices have extended instructions based on specific dtypes. For
+example, an Intel processor speeds up int8 computation using AVX
+instruction, while bool cannot benefit from that. Considering the
+complexity of dtype optimization, we add dtype as a property for
+ECG.
+Sparsity is defined as the ratio of non-zero data to all data. If data
+sparsity is relatively small, we take it as sparse data and store it in
+a compressed sparse row (CSR) format. Using sparse operators to
+handle those sparse data can perform better than dense operators.
+Taking advantage of sparsity influences optimization greatly, so we
+add sparsity as another property for ECG.
+We classify the inputs of an operator into two categories: interme-
+diate results and weights. Intermediate results are other operators’
+outputs and can only be handled during runtime. Input data is the
+first intermediate result in ECG, while output data is the last. Inter-
+mediate results are represented as {𝑠𝑝𝑎𝑟𝑠𝑖𝑡𝑦, 𝑑𝑡𝑦𝑝𝑒, 𝑡𝑒𝑛𝑠𝑜𝑟}. If we
+want to change the dtype of intermediate results, we should add
+dtype converting operator in the ECG.
+Weights are model parameters that can be loaded from trained
+models. Weights can be handled both during compilation and run-
+time, while a proper transformation during compilation can reduce
+runtime costs. Weights are represented as {𝑠𝑝𝑎𝑟𝑠𝑖𝑡𝑦, 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡_𝑑𝑡𝑦−
+𝑝𝑒,𝑎𝑐𝑡𝑢𝑎𝑙_𝑑𝑡𝑦𝑝𝑒, 𝑡𝑒𝑛𝑠𝑜𝑟}. Smallest_dtype is the smallest dtype for
+weights without accuracy loss, actual_dtype is the dtype actually
+used. Smallest_dtype depends on the property of weights, while
+actual_dtype is fixed based on smallest_dtype and operators. As
+shown in Fig. 3, 𝑊1 represents the relationship between input fea-
+tures and internal nodes for decision trees, which is a 0-1 matrix.
+The smallest_dtype of 𝑊1 is bool. However, W1 is multiplied by
+input data with a dtype of float32. If we choose bool as the ac-
+tual_dtype, 𝑊1 will be converted to float32 during runtime. To
+reduce the execution time in runtime, we should convert 𝑊1 to
+float32 during compilation, so we set actual_dtype as float32 rather
+than bool.
+Operators are represented in the form of {𝑤𝑒𝑖𝑔ℎ𝑡𝑠, 𝑖𝑛𝑡𝑒𝑟𝑚𝑒𝑑𝑖𝑎𝑡𝑒_
+𝑟𝑒𝑠𝑢𝑙𝑡𝑠, 𝑢𝑠𝑒_𝑠𝑝𝑎𝑟𝑠𝑒, 𝑡𝑦𝑝𝑒, 𝑑𝑡𝑦𝑝𝑒, 𝐷𝐿_𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟}. Weights and in-
+termediate_results are inputs of operators. Use_sparse is a flag of
+whether using the sparse operator or not, which is closely related
+to sparse operator replacing optimization described in Section 4.3.2.
+Operator type is the type of operator. As shown in Table 3, we
+Table 4: Supported Algorithms
+Preprocessing Algorithms
+Binarizer, LabelBinarizer, Normalizer, MaxAbsScaler,
+MinMaxScaler, StandardScaler, RobustScaler,
+PolynomialFeatures, LabelEncoder
+Feature Selectors
+SelectKBest, VarianceThreshold
+Linear Models
+LogisticRegression, LogisticRegressionCV, Perception,
+RidgeClassifier, RidgeClassifierCV, SGDClassifier,
+LinearRegression, Ridge, RidgeCV, SGDRegressor
+Tree-based Models
+DecisionTreeClassifier, DecisionTreeRegressor,
+ExtraTreeClassifier, ExtraTreeRegressor,
+RandomForestClassifier, RandomForestRegressor,
+ExtraTreesClassifier, ExtraTreesRegressor,
+GradientBoostingClassifier, GradientBoostingRegressor
+Support Vector Machines
+LinearSVC, LinearSVR, NuSVR, SVR
+divide operators used in ECG into five categories. Comparison op-
+erators refer to those operators that compare two tensors and return
+bool tensors. Indices operators refer to those operators that return
+tensors’ indices based on specific conditions. Those two kinds of
+operators are dtype-lowering operators, the output dtype of which
+is smaller than the input. Models without those operators, such as
+most DL models, use the same dtype through the whole graphs,
+where dtype optimizations cannot be used without approximate op-
+timization. CML models make much use of those operators, which
+have wide usage of dtype rewriting optimization described in Sec-
+tion 4.3.1. Monotonic operators refer to those operators who meet
+the following conditions:
+∀𝑥1 ≤ 𝑥2 =⇒ 𝑓 (𝑥1) ≤ 𝑓 (𝑥2)
+A series of monotonic operators followed by an indices operator
+is mathematically equivalent to the indices operators alone. Those
+properties provide more optimizations, as described in Section 4.3.3.
+Reduction operators calculate aggregates over input. Arithmetic
+operators refer to other arithmetic calculations. Operator dtype is
+the operators’ data type, such as int8 matmul or float32 matmul.
+Operator dtype depends on the dtype of weights and intermedi-
+ate_results. DL_operator is the native definition of operators in
+DL computational graphs, which we use to translate ECG to DL
+computational graphs.
+3.3
+Supported Algorithms and Extension for
+Other Algorithms
+CMLCompiler supports 35 CML algorithms nowadays, as shown
+in Table 4, covering most of the popular CML algorithms [34]. Our
+work can also be extended to other algorithms, such as clustering
+and matrix decomposition. Most CML algorithms use operators
+categorized in Section 3.1.1, each of which can be converted to cor-
+responding Operator Representations—our low-level abstractions,
+guaranteeing our extensibility. We take Kmeans as an example.
+5
+
+Xu Wen et al.
+Operator Converter
+CMLCompiler
+Model Parser
+Graph Optimizer
+Operator Representation
+Extended Computational Graph
+Optimized ECG
+Unified Abstractions
+Graph Translator
+Figure 4: The CMLCompiler architecture.
+Kmeans use basic arithmetic operators to calculate the distance
+between nodes, which can be converted to element-wise arithmetic
+operators and use aggregation operators to make clustering, which
+can be converted to reduction operators. When all operators of a
+CML algorithm are converted to Operator Representations, it can
+utilize our work to compile and make optimizations.
+4
+DESIGN AND IMPLEMENTATION
+This section illustrates the design and implementation of CMLCom-
+piler, as shown in Fig. 4. We build our framework based on the
+two unified abstractions, including four parts. Operator Converter
+converts CML operators into operator representations, as shown in
+Section 4.1. Model Parser organizes those operator representations
+in an optimization-friendly way and uses ECGs to represent CML
+models, as shown in Section 4.2. Graph Optimizer makes graph
+level optimizations, as described in Section 4.3. An optimized ECG
+is converted into a DL computational graph by Graph Translator
+in Section 4.4. DL frameworks or compilers take DL computational
+graphs as input and make more optimizations, compiling them into
+executable modules to deploy. Section 4.5 shows the mixture usage
+of CML and DL. Section 4.6 shows the implementation details.
+4.1
+Operator Converter
+Operator Converter traverses the operators in CML models and
+converts them into operator representations, respectively. Opera-
+tors based on matrices and arrays are converted into DL operators
+directly. Scalar-based operators are converted into DL operators
+based on their categories, according to Section 3.1. These converted
+DL operators are wrapped into operator representations.
+4.2
+Model Parser
+Model Parser converts operator representations into an ECG, as
+shown in Algorithm 1. Operators in an operator representation are
+initialized as nodes in an ECG, the data structure of which is defined
+in Section 3.2. Operator.weights and operator.intermediate_results
+are set according to data dependencies, and edges are built be-
+tween nodes. Operator.use_sparse and operator.dtype are set as
+False and Unknown, respectively. Operator.type is set according to
+operator type, which is defined in Table 3. Then weights and inter-
+mediate_result are initialized. Weight.sparsity is set as the ratio of
+non-zero data and all data for weight, known during compilation.
+Weight.smallest_dtype is set as the smallest dtype without accuracy
+loss, and weight.actual_dtype is initialized the same. Intermedi-
+ate_result.sparsity and intermediate_result.dtype are set according
+to operator. When all operators are visited, the ECG is established.
+Algorithm 1 Model Parser
+Input: Operator Representation
+Output: Extended Computational Graph 𝐸𝐶𝐺
+for operator in Operator Representation do
+Initialize operator as ECG node
+Set operator.weights and operator. intermediate_results ac-
+cording to data dependencies and build edges between nodes
+operator.use_sparse ← False
+operator.type ← operator type
+operator.dtype ← Unknown
+for weight in operator.weights do
+weight.sparsity ← the ratio of non-zero data and all data
+weight.smallest_dtype ← the smallest dtype without accu-
+racy loss
+weight.actual_dtype ← weight.smallest_dtype
+end for
+for ir in operator.intermediate_results do
+set ir.sparsity and ir.dtype according to operator
+end for
+end for
+4.3
+Graph Optimizer
+Graph Optimizer performs graph-level optimizations, using a func-
+tionally equivalent transformation for ECGs. These optimizations
+are based on the features of CML models and do not influence accu-
+racy. There are three specific graph rewriting optimizations: dtype
+rewriting, sparse operator replacing, and redundant elimination.
+4.3.1
+Dtype rewriting. Dtype rewriting uses low precision compu-
+tation with faster speed and less memory to replace high precision
+computation. As analyzed in Section 3.1.3, many weights used in
+CML can be represented as bool or int8. Besides, comparison opera-
+tors and indices operators widely used in CML are dtype-lowering
+operators. The intermediate results after those operators are bool or
+int8. When intermediate data and weights can be both expressed as
+low precision dtype, the corresponding operators can be converted
+into low precision computation as well.
+As shown in Fig. 5a, the top is the ECG of decision trees before
+optimization; many details are hidden. Weight 𝑊3 represents the
+relationship between leaf nodes and internal nodes for decision
+trees, which is a matrix only containing 0 and 1. The smallest_dtype
+of 𝑊3 is bool. The output of 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 operator has a dtype of bool
+as well. So the following matrix multiplication (matmul) operator
+can use a dtype of bool rather than float32. Intel processors speed
+up int8 computation using AVX instruction, while bool cannot
+benefit from that feature. So we convert the dtype of matmul to
+int8 according to hardware specification. In Fig. 5a, the below is
+the ECG after graph rewriting. Those white weights and operators
+use float32, while gray weights and operators use int8.
+6
+
+CMLCompiler: A Unified Compiler for Classical Machine Learning
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+float32
+int8
+(a) Dtype Rewriting
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+matmul
+greater
+matmul
+argmax
+W1
+W2
+W3
+input
+out
+dense
+sparse
+(b) Sparse Operator Replacing
+matmul
+add
+softmax
+argmax
+W1
+W2
+input
+out
+matmul
+add
+argmax
+W1
+W2
+input
+out
+redundant operator
+(c) Redundant Elimination
+Figure 5: Graph rewriting optimizations. Dtype rewriting converts float32 operators and weights into low-precision. Sparse
+operator replacing converts dense operators and weights into sparse. Redundant elimination reduces redundant operators.
+Now we introduce the dtype rewriting principle in detail. Algo-
+rithm 2 shows the procedure of dtype rewriting:
+(1) Visit all operators in ECG. For each operator, dtype is set as the
+largest dtype of all inputs. After that, operator dtype is converted to
+the dtype which can utilize hardware’s SIMD instructions best. We
+keep a list of hardware specifications to modulate operator dtype.
+In order to guarantee accuracy, dtype cannot get smaller. Then we
+modulate operator implementation based on operator dtype.
+(2) When operator dtype is fixed, we set the input dtype. The
+dtype of weights is set the same as the operator, reducing dtype
+conversion in runtime. The dtype of intermediate results cannot be
+converted during compilation. So we add dtype converting operator,
+.i.e, cast, before the operator.
+We explain the differences between dtype rewriting for CML
+models and model quantization for DL models. Quantization is an
+approximate algorithm for DL models that causes a decrease in
+accuracy and brings extra computation, such as calibration. Dtype
+rewriting for CML models is based on the properties of CML, con-
+verting dtype of operators and weights with no accuracy decrease
+and extra computation.
+Algorithm 2 Dtype Rewriting
+Input: ECG 𝐺, hardware configuration 𝐻
+Output: Optimized ECG 𝐺′
+for operator in 𝐺 do
+operator.dtype ← largest dtype in operator.weights and oper-
+ator.intermediate_results
+Modulate operator.dtype based on 𝐻
+Modulate operator.DL_operator based on operator.dtype
+for weight in operator.weights do
+weight.actual_dtype ← operator.dtype
+end for
+for data in operator.intermediate_results do
+if data.dtype < operator.dtype then
+Add cast(data, operator.dtype) before operator
+end if
+end for
+end for
+4.3.2
+Sparse operator replacing. Replacing dense operators with
+sparse operations can speed up as well. Algorithm 3 shows the
+procedure of sparse operator replacing. The sparsity of input data
+can be known until runtime, while the sparsity of weights can
+be known during compilation. So we convert the data format of
+weights rather than input data. Different hardware devices have
+different support for sparse operators. For example, CPUs can ben-
+efit from sparse computation while GPUs have little effect. So we
+set a threshold based on hardware specification. If weight sparsity
+is smaller than the threshold, we store it in a compressed sparse
+row (CSR) format. Then we convert the corresponding operator
+into a sparse implementation. An example is shown in Fig. 5b, we
+convert 𝑊1 and the corresponding matmul to sparse.
+Algorithm 3 Sparse Operator Replacing
+Input: ECG 𝐺, Threshold 𝑇
+Output: Optimized ECG 𝐺′
+for operator in 𝐺 do
+for weight in operator.weights do
+if weight.sparsity < 𝑇 then
+Store weight into CSR format
+operator.use_sparse ← True
+Convert operator.DL_operator into sparse implementa-
+tion
+end if
+end for
+end for
+4.3.3
+Redundant elimination. Redundant elimination eliminates
+those operators who do not influence final results due to their math-
+ematical properties. For example, a series of monotonic operators
+followed by an indices operator is mathematically equivalent to
+the indices operators alone. Algorithm 4 shows the procedure of
+redundant elimination. For each operator in ECGs, we check its
+operator type. If another monotonic operator follows a monotonic
+operator, we fuse them. We eliminate the monotonic operator if it
+is followed by an indices operator. An example is shown in Fig. 5c,
+the softmax before argmax is eliminated.
+4.4
+Graph Translator
+Graph Translator converts the optimized ECG into DL computa-
+tional graph, choosing the proper implementation based on ECG
+7
+
+Xu Wen et al.
+Algorithm 4 Redundant Elimination
+Input: Extended Computational Graph 𝐺
+Output: Optimized ECG 𝐺′
+for operator in 𝐺 do
+if operator.type == "monotonic" then
+Check the next operator operator’
+if operator’.type == "monotonic" then
+Merge operator and operator’
+else if operator’.type == "indices" then
+Eliminate operator
+end if
+end if
+end for
+DL models
+CML models
+Single ECG for hybrid models
+Cross-framework implementation
+Figure 6: CMLCompiler uses a single ECG to represent CML
+and DL mixed pipeline.
+and hardware specification information. DL frameworks or compil-
+ers, like TVM, take DL computational graphs as input and make
+more optimizations, finally compiling them into executable mod-
+ules.
+4.5
+Hybrid Deployment of CML and DL with a
+Unified Framework
+We convert those CML and DL hybrid applications under a unified
+framework to reduce the cost of switching frameworks and provide
+an opportunity for end-to-end optimizations, as shown in Fig. 6. We
+load models from PyTorch and sklearn and convert them into ECG
+subgraphs. We build edges according to data dependency and merge
+those subgraphs in a single ECG. Then we can use optimizations
+both in our work and DL compilers. Finally, we compile and deploy
+it on diverse hardware devices.
+4.6
+Implementation
+Due to the benefits in portability and performance, we implement
+CMLCompiler on the basis of TVM. The intermediate representa-
+tions and transforms are all written in python. We read trained
+models from CML frameworks such as sklearn and convert them
+into operator representations, implementing them in the format
+of TVM relay functions and storing their weights in TVM arrays.
+We wrap those relay functions in the format of ECGs. After opti-
+mizations in Section 4.3, we convert ECGs into TVM’s IRModules.
+Then we utilize TVM to make more optimizations and compile to
+executable modules based on specific hardware targets. We use
+cross-compilation to support a broad spectrum of hardware devices.
+We deploy them on lightweight runtime based on TVM runtime
+and make inference on various hardware devices.
+5
+EVALUATION
+This section summarizes the evaluation. Section 5.1 shows experi-
+mental setup. Section 5.2 evaluates the performance of graph rewrit-
+ing optimizations based on ECGs. Section 5.3 compares our work
+with the state-of-the-art frameworks. Section 5.4 evaluates the hy-
+brid deployment of CML and DL.
+5.1
+Experimental Setup
+We deploy a server node equipped with two Xeon E5-2620 V3
+(Haswell) CPUs, an Nvidia Titan RTX GPU, and 64 GB memory to
+conduct the experiments on CPU and GPU. Each CPU contains six
+physical cores. The GPU contains 4608 Cuda cores and 24 GB mem-
+ory. The operating system is Ubuntu 16.04, and the other software
+includes TVM 0.8, PyTorch 1.8.1, hummingbird 0.3.1, scikit-learn
+1.0.1, and CUDA 10.2. For the IoT experiments, we use Raspber-
+rypi4b with Raspbian 10 operating system and deploy the above
+software with the same version. We use YearPrediction [12] as the
+dataset, with 515345 samples and 90 features. We use 80% data to
+train models and 20% data to make inference. We run all the exper-
+iments five times and use the average as the final results. We test
+hummingbird [30] using both two backends (PyTorch and TVM)
+and select their best results.
+5.2
+Optimizations
+This section evaluates graph rewriting optimizations based on
+ECGs, as described in Section 4.3. These optimizations: dtype rewrit-
+ing, sparse operator replacing, and redundant elimination, can work
+together and produce cumulative optimization effects. They can
+also coexist with the optimizations in TVM. We choose four typ-
+ical tree models: DecisionTreeClassifier, RandomForestClassifier,
+ExtraTreeClassifier, and ExtraTreesClassifier, as well as two typical
+linear models: LogisticRegression and SGDClassifier. We evaluate
+the dtype rewriting and sparse operator replacing for tree models,
+and redundant elimination for linear models according to their
+unique patterns.
+Fig. 7a shows the result on CPU. For tree models, using our work
+without optimizations has a 1.31x-2.54x speedup compared with
+sklearn; this is due to our abstractions which utilize optimizations
+of TVM, including better utilization of SIMD instructions and multi
+cores. Using dtype rewriting and sparse operator replacing bring
+1x-1.21x and 1.26x-1.75x speedup, respectively, achieving 1.27x-
+2.11x speedup together, 1.84x-4.44x faster than sklearn. For linear
+models, our work without optimizations runs slower than sklearn.
+However, using redundant elimination brings 1.22x-1.51x speedup;
+the result after our optimizations is 1.06x-1.14x faster than sklearn.
+Fig. 7b shows the result of IoT devices. Note that sklearn lacks
+enough support for IoT devices. For example, 64-bit tree models
+trained on servers cannot be executed on Raspberrypi4b with a
+32-bit operating system. Retraining those models in 32-bit format
+on Raspberrypi4b from scratch takes more time, so we regard those
+models as unsupported, marked as cross. So we take our work with-
+out optimizations as the baseline. Using dtype rewriting and sparse
+operator replacing bring 1.01x-1.33x and 1.23x-2.3x speedup, respec-
+tively, achieving 1.49x-2.53x speedup together. For linear models,
+8
+
+CMLCompiler: A Unified Compiler for Classical Machine Learning
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+(b) Raspberrypi4b
+Figure 7: Graph Rewriting Optimizations. "base" means our work without optimizations. "DR" means only using dtype rewrit-
+ing. "DR+SOR" means using both dtype rewriting and sparse operator replacing. "RE" means using redundant elimination.
+our work without optimizations achieves 1.71x-1.84x speedup. Us-
+ing redundant elimination brings 1.08x-1.14x more speedup, 1.95x-
+1.98x faster than sklearn. The computation part of GPU is less than
+20%, so those optimizations play a limited role on GPU. In conclu-
+sion, CML models can benefit from both TVM’s optimizations and
+our optimizations and achieve obvious speedup.
+5.3
+Overall Results
+This section evaluates 14 typical CML algorithms covering prepro-
+cessing algorithms, linear models, tree-based models, and SVMs,
+on CPU, GPU, and IoT devices, compared with state-of-the-art
+frameworks including sklearn, intel extension for sklearn [20], and
+hummingbird. It contains two parts: batch experiments for all data
+and query experiments for a single record.
+The differences between the accuracy of CMLCompiler and
+sklearn are all less than 1 × 10−5, which means that our work
+does not affect the accuracy. The outputs on different hardware
+are all the same, so we focus on performance hereinafter. Table 5
+shows the performance of batch experiments. On CPU, our work
+reflects the best performance on 12 algorithms out of 14, achieving
+1.02x-10.57x speedup compared with sklearn, 1.14x-4.38x speedup
+compared with hummingbird, and 1.44x-8.47x speedup compared
+with intel sklearn. On GPU, our work achieves competitive perfor-
+mance compared with hummingbird. Our work performs better
+on 11 algorithms out of 14, with a 1.11x-3.31x speedup. On an IoT
+device Raspberrypi4b, our work performs better on 13 algorithms
+out of 14, with a 1.28x-5.09x speedup.
+Table 6 shows the performance of query experiments for a single
+record. On CPU, our work achieves the best performance on 11
+algorithms out of 14, with a 1.36x-170.68x speedup compared with
+sklearn, a 1.56x-4.47x speedup compared with hummingbird, and
+a 1.31x-169.43x speedup compared with intel sklearn. Our work
+has better performance on GPU on 10 algorithms out of 14 com-
+pared with hummingbird, with a 1.41x-4.64x speedup. Our latency
+on Raspberrypi4b does not differ much compared with sklearn.
+However, we perform better in model support.
+In conclusion, we have advantages in both batch and query ex-
+periments for all three hardware devices. Many models in sklearn
+only support a single core and cannot fully utilize the SIMD in-
+structions. We perform better than sklearn and intel sklearn due
+to better utilization of multi cores and SIMD instructions through
+compilation. Hummingbird uses both PyTorch and TVM as back-
+ends, where TVM performs better in most cases of our evaluations.
+It implements models in PyTorch and converts them into TVM
+using 𝑓 𝑟𝑜𝑚_𝑝𝑦𝑡𝑜𝑟𝑐ℎ API. This conversion is not direct and effi-
+cient enough, causing a performance decrease. Besides, hardware
+information is missed during conversion, which limits the optimiza-
+tions of TVM for hummingbird. We map ECGs into relay opera-
+tors directly and select the most efficient implementation based on
+ECGs and hardware specification information. Additionally, our
+abstractions bring more optimizations, as described in Section 4.3,
+bringing up to 2.53x speedup, working together to achieve better
+performance.
+5.4
+Hybrid Deployment of CML and DL
+This section shows three hybrid deployment cases of CML and DL.
+As the baselines, without a unified framework, a DL framework
+is used to implement DL algorithms, while a CML framework is
+used to implement CML algorithms. Our work converts CML and
+DL models into a single ECG, making optimizations and compiling
+to diverse hardware devices. We test the latency of a single query,
+which is essential in real-world applications.
+5.4.1
+Sentence Sentiment Classification. The first one is a sentence
+sentiment classification case, which uses Bert to embed English
+sentences and logistic regression to make a classification [36]. We
+use BERT-tiny [3] as the pre-trained Bert model and SST2 [40]
+as the dataset. The baseline implements BERT-tiny in pytorch-
+transformers [45] and logistic regression in sklearn. The result is
+shown in Fig .8a. Our work achieves a 1.67x speedup on server
+CPUs. Pytorch-transformers cannot be installed on IoT devices, so
+the baseline cannot run on Raspberrypi4b. The latency of our work
+on Raspberrypi4b is 18 milliseconds, which is acceptable in most
+use cases.
+5.4.2
+Radiographic Image Analysis. The second case uses Deep
+Hybrid Learning [38] to analyze radiographic images, which uses
+9
+
+Xu Wen et al.
+Table 5: Execution time for batch experiments over all data on CPU (12 cores), GPU, and IoT devices (take Raspberrypi4b
+as an example) in milliseconds. SK, HB, and Intel is short for scikit-learn, hummingbird, and Intel extension for sklearn,
+respectively. "-" means unsupported.
+Algorithm
+CPU
+GPU
+IOT
+SK
+HB
+Intel
+Our
+HB
+Our
+SK
+Our
+Binarizer
+97
+31
+77
+9
+19
+6
+634
+126
+Normalizer
+25
+33
+15
+15
+7
+5
+241
+168
+MinMaxScaler
+19
+31
+13
+8
+21
+6
+199
+148
+RobustScaler
+28
+32
+25
+12
+19
+5
+343
+156
+LinearRegression
+12
+18
+4
+6
+6
+7
+61
+116
+LogisticRegression
+98
+104
+137
+86
+7
+7
+1889
+952
+SGDClassifier
+94
+98
+139
+88
+9
+7
+1886
+969
+DecisionTreeClassifier
+33
+48
+23
+16
+7
+5
+-
+99
+DecisionTreeRegressor
+7
+19
+3
+15
+7
+6
+-
+211
+RandomForestClassifier
+2130
+885
+2003
+601
+20
+-
+-
+5820
+ExtraTreeClassifier
+29
+-
+26
+16
+-
+6
+-
+206
+ExtraTreesClassifier
+10022
+2522
+9421
+2256
+99
+-
+-
+47959
+LinearSVC
+92
+122
+152
+77
+9
+6
+1896
+930
+LinearSVR
+39
+26
+34
+5
+6
+5
+323
+112
+Table 6: Latency for query experiments over one single record on CPU (12 cores), GPU, and IoT devices (take Raspberrypi4b
+as an example) in milliseconds. The symbols are the same as Table 5.
+Algorithm
+CPU
+GPU
+IOT
+SK
+HB
+Intel
+Our
+HB
+Our
+SK
+Our
+Binarizer
+0.2
+0.26
+0.34
+0.09
+0.93
+0.64
+0.44
+0.59
+Normalizer
+0.32
+0.26
+0.28
+0.11
+0.25
+0.68
+0.59
+0.41
+MinMaxScaler
+0.15
+0.31
+0.14
+0.09
+0.91
+0.63
+0.33
+0.37
+RobustScaler
+0.14
+0.22
+0.14
+0.11
+1.02
+0.72
+0.37
+0.37
+LinearRegression
+0.24
+0.35
+0.32
+0.1
+0.91
+0.55
+0.52
+0.69
+LogisticRegression
+0.35
+0.36
+0.29
+0.19
+3.29
+0.71
+0.67
+2.59
+SGDClassifier
+0.4
+0.35
+0.29
+0.23
+2.93
+0.67
+0.68
+0.65
+DecisionTreeClassifier
+0.24
+1.62
+0.27
+0.36
+3.01
+0.8
+-
+0.9
+DecisionTreeRegressor
+0.22
+0.22
+0.25
+0.38
+1.03
+0.72
+-
+0.88
+RandomForestClassifier
+103.96
+1.6
+103.2
+0.61
+2.56
+-
+-
+1.05
+ExtraTreeClassifier
+0.23
+-
+0.4
+0.47
+-
+-
+-
+1.81
+ExtraTreesClassifier
+205.27
+12.74
+204.25
+1.73
+2.41
+-
+-
+3.11
+LinearSVC
+0.4
+0.37
+0.45
+0.19
+2.71
+0.61
+0.65
+1.07
+LinearSVR
+0.31
+0.34
+0.37
+0.09
+0.91
+0.62
+0.54
+0.91
+���
+�������������
+�
+�
+��
+��
+������������
+��������
+��������
+(a) Bert+LogisticRegression for sentence
+sentiment classification
+���
+�������������
+�
+��
+��
+��
+������������
+��������
+��������
+(b) SimpleDNN+RandomForest for
+radiographic image analysis
+���
+�������������
+�
+�
+��
+��
+������������
+��������
+��������
+(c) GBDT+Wide&Deep for click through
+prediction
+Figure 8: The latency of a single query for CML and DL mixed pipelines. All three baselines cannot run on IoT devices.
+10
+
+CMLCompiler: A Unified Compiler for Classical Machine Learning
+simple DNN to make feature engineering and CML models such as
+random forests to make a classification. We use CheXpert [21] as
+the dataset. The baseline implements DNN in PyTorch and random
+forest in sklearn. The result is shown in Fig .8b. Our work achieves a
+2.3x speedup on server CPUs. The pre-trained random forest cannot
+run on IoT devices, while our work solves this problem through
+cross-compilation.
+5.4.3
+Click Through Rate Prediction. The third case is click-through
+rate prediction used in recommendation systems of our anonymous
+industry partners, using GBDT [15] to extract features and the
+Wide and Deep [9] models to make prediction. We use avazu 1 as
+the dataset. The baseline implements GBDT in sklearn and Wide
+and Deep in PyTorch. The result is shown in Fig .8c. We achieve
+3.04x speedup on the server CPUs. The GBDT model in the baseline
+cannot be executed on IoT devices, while our latency on IoT devices
+is only 5.06 ms.
+6
+RELATED WORK
+CML frameworks and libraries can be divided into three categories.
+(1) General-purpose solution uses one framework to support various
+models. Scikit-learn [32] is the most widely used CML framework
+on GitHub [33]. Spark MLlib [29] is an extension to Spark [48].
+H2O [17] uses MapReduce [11] to support both CML and DL. There
+are many other works, such as Shogun [41] and RapidMiner [19].
+These frameworks only support CPU, suffering from severe perfor-
+mance and portability issues. (2) Specific-purpose solution focuses
+on one type of model. LibLinear [14] supports logistic regression
+and linear SVM. LibSVM [5] focuses on SVMs. These works are
+limited to CPUs. Some other works attempt to support various hard-
+ware devices. XGBoost [6] implements gradient boosting decision
+tree algorithm on CPUs and GPUs. Muhsen Owaida et al. [31] bring
+XGBoost to FPGAs. Toby Sharp [39] implements decision trees and
+forests on GPUs. These frameworks only support a narrowed vari-
+ety of models and solve the problem of portability to a certain extent.
+(3) Extension based on DL attempts to utilize DL frameworks to
+support CML models. TF-DF [43] is a decision forest library based
+on TensorFlow but is limited to CPUs. It’s implemented in an ad-hoc
+way, losing the portability of DL frameworks. Hummingbird [30]
+is a general-purpose solution based on PyTorch, adding support
+for GPUs. They utilize those abstractions in DL frameworks di-
+rectly without digging into the features of CML, missing many
+optimization chances.
+7
+CONCLUSION
+This paper presented the design and implementation of CMLCom-
+piler, a unified compiler for classical Machine Learning (CML) in-
+ference. CMLCompiler proposed two unified abstractions: oper-
+ator representations and extended computational graphs (ECGs).
+Operator representations convert CML operators into tensor for-
+mats, while an ECG organizes these converted operators in an
+optimization-friendly way. The CMLCompiler framework performs
+the conversion and graph optimization based on two unified ab-
+stractions, then outputs an optimized computational graph to deep
+learning compilers or frameworks. CMLCompiler also enables the
+1https://www.kaggle.com/c/avazu-ctr-prediction
+hybrid deployment of CML and DL with a unified framework. Our
+implementations of CMLCompiler on top of TVM show the effec-
+tiveness and achieve up to 4.38x speedup on CPU, 3.31x speedup
+on GPU, and 5.09x speedup on IoT devices, compared to the state-
+of-the-art solutions — scikit-learn, intel sklearn, and hummingbird.
+Our support for CML and DL mixed pipelines achieves up to 3.04x
+speedup compared with cross-framework implementations.
+A
+PROOF
+Here we prove that 𝑎𝑟𝑔𝑚𝑎𝑥 in Fig. 3 returns the leaf node finally
+reaches. 𝑁𝑆, 𝑁𝐼, and 𝑁𝐿 refer to the number of samples, internal
+nodes, and leaf nodes, respectively. 𝐼 refers to internal nodes, num-
+bered in the order of Level Order Traversal. 𝐿 refers to leaf nodes,
+numbered in the order of In-Order Traversal. 𝑋 ∈ {0, 1}𝑁𝑆×𝑁𝐼 is
+the result after comparison with 𝑊2. Each row 𝑋𝑖 ∈ {0, 1}𝑁𝐼 refers
+to choices for one sample x, marked as �𝑥. 𝑊3 ∈ {0, 1}𝑁𝐼 ×𝑁𝐿 can
+be regarded as a list of column vector { �𝐿1, �𝐿2,..., �
+𝐿𝑁𝐿}. �𝐿𝑖 ∈ {0, 1}𝑁𝐼
+represents the relationship between leaf node 𝐿𝑖 and all internal
+nodes. Then we should prove that 𝑎𝑟𝑔𝑚𝑎𝑥(�𝑥 · �𝐿1, �𝑥 · �𝐿2, ..., �𝑥 ·
+�
+𝐿𝑁𝐿)
+returns the leaf x reaches, where 𝑎𝑟𝑔𝑚𝑎𝑥 returns the index of the
+maximum values among the input tensor. It returns the first index
+if maximum appears more than once. We assume that 𝐿𝑘 is the leaf
+node x reaches.
+First we prove that �𝑥 · �𝐿𝑘 is the maximum value in {�𝑥 · �𝐿1, �𝑥 ·
+�𝐿2, ..., �𝑥 ·
+�
+𝐿𝑁𝐿 }. We define the path from root node 𝐼0 to 𝐿𝑘 as the
+decision path of x.
+𝐿𝑘 [𝑖] =
+� 0, choose left in 𝐼𝑖 and 𝐼𝑖 ∈ 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛𝑝𝑎𝑡ℎ
+1, otherwise
+𝑥[𝑖] =
+� 0, choose left in 𝐼𝑖
+1, choose right in 𝐼𝑖
+Because x reaches 𝐿𝐾, if x[i] = 1 and 𝐼𝑖 ∈ 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑝𝑎𝑡ℎ, then
+𝐿𝑘 [𝑖] = 1. DP represents decision path, right means choosing right
+in internal node and left means choosing left in internal node.
+�𝑥 · �𝐿𝑘 =
+∑︁
+𝑖
+𝑥[𝑖] ∗ 𝐿𝑘 [𝑖]
+=
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖
+1 ∗ 𝐿𝑘 [𝑖] +
+∑︁
+𝑖, 𝑙𝑒𝑓 𝑡 𝑖𝑛 𝐼𝑖
+0 ∗ 𝐿𝑘 [𝑖]
+=
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖
+1 ∗ 𝐿𝑘 [𝑖]
+=
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖 ∈𝐷𝑃
+1 ∗ 𝐿𝑘 [𝑖] +
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖∉𝐷𝑃
+1 ∗ 𝐿𝑘 [𝑖]
+=
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖 ∈𝐷𝑃
+1 ∗ 1 +
+∑︁
+𝑖, 𝑟𝑖𝑔ℎ𝑡 𝑖𝑛 𝐼𝑖∉𝐷𝑃
+1 ∗ 1
+= 𝐶𝑜𝑢𝑛𝑡𝑠 𝑜𝑓 1 𝑖𝑛 �𝑥
+�𝑥 and { �𝐿1, �𝐿2,..., �
+𝐿𝑁𝐿} are all 0-1 vector. Counts of 1 in �𝑥 is the maxi-
+mum value of {�𝑥 · �𝐿1, �𝑥 · �𝐿2, ..., �𝑥 ·
+�
+𝐿𝑁𝐿 }.
+Then we prove that 𝑘 is the first index that returns the maximum.
+We assume that there exists a leaf node 𝐿𝑡 ahead of 𝐿𝐾 which meets
+the condition �𝑥 · �𝐿𝑡 == 𝑚𝑎𝑥𝑖𝑚𝑢𝑚. Now that 𝐿𝑡 is ahead of 𝐿𝑘 and
+the leaf nodes are numbered in a In-Order Traversal. ∃ an internal
+node 𝐼𝑖 where 𝐿𝑡 is in the left subtree of 𝐼𝑖 and 𝐿𝑘 in the right subtree
+of 𝐼𝑖. X passes by 𝐼𝑖 and reaches 𝐿𝐾 in its right subtree, so x[i] = 1.
+𝐿𝑡 is in the left subtree of 𝑇𝑖, so 𝐿𝑡 [𝑖]= 0, where x[i] is multiplied
+11
+
+Xu Wen et al.
+by zero. So �𝑥 · �𝐿𝑡 < 𝑚𝑎𝑥𝑖𝑚𝑢𝑚. Conflict with the assumption that
+�𝑥 · �𝐿𝑡 == 𝑚𝑎𝑥𝑖𝑚𝑢𝑚. So 𝐿𝑘 is the first index that returns maximum.
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+Astronomy & Astrophysics manuscript no. 44705corr
+©ESO 2023
+January 9, 2023
+A framework for the architecture of exoplanetary systems
+II. Nature versus nurture: Emergent formation pathways of architecture classes
+Lokesh Mishra1, 2 , Yann Alibert1 , Stéphane Udry2
+, and Christoph Mordasini1
+1 Institute of Physics, University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
+e-mail: exomishra@gmail.com
+2 Geneva Observatory, University of Geneva, Chemin Pegasi 51b, 1290 Versoix, Switzerland
+Received DD MMM YYYY; accepted DD MMM YYYY
+ABSTRACT
+In the first paper of this series, we proposed a model-independent framework for characterising the architecture of planetary systems
+at the system level. There are four classes of planetary system architecture: similar, mixed, anti-ordered, and ordered. In this paper,
+we investigate the formation pathways leading to these four architecture classes. To understand the role of nature versus nurture in
+sculpting the final (mass) architecture of a system, we apply our architecture framework to synthetic planetary systems — formed
+via core-accretion — using the Bern model. General patterns emerge in the formation pathways of the four architecture classes.
+Almost all planetary systems emerging from protoplanetary disks whose initial solid mass was less than one Jupiter mass are similar.
+Systems emerging from heavier disks may become mixed, anti-ordered, or ordered. Increasing dynamical interactions (planet–planet,
+planet–disk) tends to shift a system’s architecture from mixed to anti-ordered to ordered. Our model predicts the existence of a new
+metallicity–architecture correlation. Similar systems have very high occurrence around low-metallicity stars. The occurrence of the
+anti-ordered and ordered classes increases with increasing metallicity. The occurrence of mixed architecture first increases and then
+decreases with increasing metallicity. In our synthetic planetary systems, the role of nature is disentangled from the role of nurture.
+Nature (or initial conditions) pre-determines whether the architecture of a system becomes similar; otherwise nurture influences
+whether a system becomes mixed, anti-ordered, or ordered. We propose the ‘Aryabhata formation scenario’ to explain some planetary
+systems which host only water-rich worlds. We finish this paper with a discussion of future observational and theoretical works that
+may support or refute the results of this paper.
+Key words. Planetary systems – Planets and satellites: detection – Planets and satellites: formation – Planets and satellites: physical
+evolution
+1. Introduction
+Studying planetary systems as single units of a physical sys-
+tem makes them amenable to system level examinations. Inves-
+tigating the ensemble of bound objects (host star(s), planets, mi-
+nor bodies) coherently can allow a deeper and more compre-
+hensive understanding of exoplanetary astrophysics to emerge.
+The purview of this multi-body physics covers a breadth of
+topics including stability of planetary systems (Gladman 1993;
+Laskar 1997, 2000; Chambers 1999; Fang & Margot 2013; Pu
+& Wu 2015; Laskar & Petit 2017; Obertas et al. 2017; Petit
+et al. 2018; Wang et al. 2019; Yeh et al. 2020; Tamayo et al.
+2020; Turrini et al. 2020), stellar host and protoplanetary disk
+properties (Petigura et al. 2018; Manara et al. 2019; Mulders
+et al. 2021), novel approaches to system-level characterisation
+(Tremaine 2015; Kipping 2018; Alibert 2019; Mishra et al. 2019;
+Gilbert & Fabrycky 2020; Bashi & Zucker 2021; Sandford et al.
+2021), and the architecture of planetary systems (Lissauer et al.
+2011; Ciardi et al. 2013; Fabrycky et al. 2014; Weiss et al.
+2018; Millholland et al. 2017; Adams 2019; Adams et al. 2020;
+Mulders et al. 2020; He et al. 2019; He et al. 2021; Mishra
+et al. 2021; Adibekyan et al. 2021; Millholland & Winn 2021;
+Winter et al. 2020). Analysing multi-body system level physics
+may allow us to understand whether planetary systems are self-
+organizing emergent structures – i.e. whether global level pat-
+terns are emerging from local level interactions.
+Inspired by the peas in a pod architecture (Weiss et al. 2018;
+Millholland et al. 2017; Mishra et al. 2021), we introduced a new
+framework for studying the architecture of planetary systems
+(Mishra et al. 2023, ; hereafter Paper I). Studying the architecture
+as a global-system level phenomena, this framework allows us to
+characterise, quantify, and compare the architecture of individual
+planetary systems. Four classes of planetary system architecture
+emerged from this framework. These classes are labelled simi-
+lar, mixed, anti-ordered, and ordered depending on the arrange-
+ment and distribution of planets around the host star. The key
+idea behind this framework is that the arrangement and distribu-
+tion of planets contains additional information that cannot be ex-
+tracted by studying single planets individually. Hints of the pres-
+ence of this additional information were revealed in some works
+(Tremaine 2015; Laskar & Petit 2017; Kipping 2018; Mishra
+et al. 2019; Gilbert & Fabrycky 2020; Sandford et al. 2021).
+Explaining the formation, evolution, and final assembly of
+planetary systems remains an outstanding theoretical problem.
+Planet-formation physics spans astronomical orders of magni-
+tude in mass, size, and time (Udry & Santos 2007; Armitage
+2010). The processes occurring during planet formation convert
+gases and micron-sized dust particles from the protoplanetary
+disk into different kinds of planets arranged in different architec-
+tures over timescales of millions and billions of years. However,
+it remains unclear as to how initial conditions derived from the
+host star or protoplanetary disk combine with the formation and
+Article number, page 1 of 12
+arXiv:2301.02373v1  [astro-ph.EP]  6 Jan 2023
+
+A&A proofs: manuscript no. 44705corr
+evolution processes to give rise to the observed exoplanetary sys-
+tems.
+We are interested in understanding the role of nature ver-
+sus nurture in sculpting the final planetary system and the extent
+to which the character of the mature planetary system is influ-
+enced by its initial conditions. Kipping (2018) suggested, using
+an entropy-like formulation for planetary systems, that the ini-
+tial conditions of planet formation could be inferred based on
+their present-day architecture. However, the presence of stochas-
+tic processes makes it difficult to connect the initial conditions
+with the final system. It is also unclear as to whether or not
+stochastic physical processes can erase all memory of initial con-
+ditions, or indeed leave their own impressions on the final archi-
+tecture. Using ideas from the fields of machine-learning-based
+natural language processing, Sandford et al. (2021) showed that
+planetary systems are not randomly assembled. While it is clear
+that planetary systems are not identical copies of one another, the
+quest for quantifying the similarity between planetary systems is
+a tantalising one.
+In this paper, we investigate the formation pathways that
+lead to the four architecture classes. Due to the stochastic na-
+ture of this problem, understanding the formation of a single
+planetary system can be very complicated. For example, two
+systems with almost identical initial conditions may evolve into
+two completely different planetary systems. Chaos arising from
+multi-body gravitational interactions may cause differing forma-
+tion pathways for these two systems. However, some patterns
+are found to emerge when studying planetary systems as part of
+an ensemble. These trends, as we show in this paper, help us
+to understand the role played by initial conditions and physical
+processes in shaping the architecture.
+Figure 1 (bottom) summarises the main findings of this pa-
+per. We show that the effects of planet formation and evolution
+processes are imprinted in the system-level architecture. Fig-
+ure 1 shows the formation pathways of the architecture classes
+that emerge due to the system-level approach of our architecture
+framework (Fig. 1 (top)). This sankey diagram has nodes for pro-
+toplanetary disk gas mass, protoplanetary disk solid mass, metal-
+licity, and planetary architecture. We find that the formation of
+similar planetary systems is dominated by initial conditions. If
+the initial conditions disfavour the formation of similar archi-
+tecture, the other three architectures may emerge. Whether the
+final architecture is mixed, ordered, or anti-ordered seems to de-
+pend on the stochastic formation processes. Increasing dynam-
+ical interactions (disk–planet, planet–planet) generally tends to
+produce mixed, anti-ordered, and then ordered architectures, re-
+spectively.
+We first summarise the architecture framework and some re-
+sults from Paper I in Sect. 2. We study the role of nature (initial
+conditions) and nurture (dynamical processes) in Sects. 3 and
+4, respectively. In these sections, we study the influence of pro-
+toplanetary disk mass, metallicity, protoplanetary disk lifetime,
+planet–disk interactions, planet–planet interactions, and N-body
+interactions on the final architecture of simulated planetary sys-
+tems. We summarise our results, suggest possible future studies,
+and conclude this paper in Sect. 6.
+2. Summary of Paper I and the Bern model
+2.1. Architecture framework
+The arrangement of multiple planets and the collective distri-
+bution of their physical properties around the host star(s) char-
+acterises the architecture of a planetary system (Mishra et al.
+Distance from star
+Quantity (e.g. Mass)
+Similar
+Anti-Ordered
+Ordered
+Mixed
+Fig. 1. The four classes of planetary system architecture and their emer-
+gent formation pathways.
+Top: Reproduced from Paper I – Schematic diagram depicting the Four
+classes of planetary system architecture: similar, anti-ordered, mixed,
+and ordered. Depending on how a quantity (such as mass or size) varies
+from one planet to another, the architecture of a system can be identi-
+fied. The framework is model independent.
+Bottom: Emergence of formation pathways: Sankey diagram depicting
+the emergence of formation pathways of architecture classes. The thick-
+ness of the links and nodes is proportional to the relative number of
+synthetic systems in our simulation. This result is derived from syn-
+thetic planetary systems around a solar mass star via the Bern model.
+Disk gas mass and metallicity are binned at their median values.
+2021). To quantify the architecture of a planetary system, we de-
+veloped a novel model-independent framework Paper I. Some
+key aspects of this framework are briefly summarised here, and
+we refer the reader to Sect. 3 of Paper I for details.
+Conceptually, the framework defines four classes of plane-
+tary system architecture: similar, mixed, anti-ordered, and or-
+Article number, page 2 of 12
+
+Protoplanetary Disk
+Mgas
+< 0.03Mo
+disk
+Gas
+Star Metallicity
+[Fe/H] < 0
+[Fe/H] ≥0
+Protoplanetary Disk
+≥
+1MJ
+Solids
+Planetary System
+Architecture Class
+Ordered
+Similar
+Ordered
+Mixed
+AntiL. Mishra et al.: Architecture Framework II – Nature versus nurture: Emergent formation pathways of architecture classes
+dered. Consider a planetary quantity (such as mass, radius, etc.)
+as a function of the distance of the planet to the host star (see Fig.
+1). When all planets in a system have similar values of a plane-
+tary quantity, the architecture of such systems is similar. When
+the planetary quantity increases with increasing distance, the
+system is said to exhibit an ordered architecture. Alternatively,
+if the quantity shows an overall decreasing trend with increas-
+ing distance, the architecture is considered to be anti-ordered.
+Finally, the planetary quantities could also show variations that
+are not captured in the three classes above. A mixed architec-
+ture may depict large, increasing, and decreasing variations with
+distance. By studying the variation of a planetary quantity with
+distance for all planets in the system, our framework captures the
+arrangement and distribution of planets in the system.
+The architecture of a system is quantified via two coeffi-
+cients: the coefficient of similarity, CS (qi), and the coefficient of
+variation, CV(qi). Here, qi represents a planetary quantity (e.g.
+mass, radius, eccentricity, density) for the ith planet. When the
+coefficients are calculated using planetary masses, they inform
+us about the mass architecture of a system, that is, the arrange-
+ment and distribution of mass in a given system. Likewise, we
+can study the radius architecture, density architecture, water-
+mass-fraction architecture, eccentricity architecture, and so on.
+The versatility of our architecture framework lies in its ability
+to allow us to study the multifaceted architectures of a planetary
+system. In Paper I, we explored the relationship between these
+different kinds of architectures. As in Paper I, we identify the
+architecture of a system by its bulk mass architecture.
+Calibrated on planetary masses, a classification scheme to
+identify the architecture class was proposed in Paper I (eq. 8).
+The CS versus CV plane represents the architecture space for
+planetary systems (Fig. 3 in Paper I). This new parameter space
+was found to be endowed with a curious mathematical property,
+namely planetary systems cannot occupy all parts of the archi-
+tecture plane, as some regions of this parameter space are math-
+ematically forbidden.
+To understand the implications of this architecture frame-
+work, we applied it on several catalogues in Paper I. These
+included 41 observed multi-planetary systems and numerically
+simulated systems via population synthesis using the Generation
+III Bern model (Emsenhuber et al. 2021a,b).
+2.2. Bern model
+For the synthetic planetary systems, as the initial conditions and
+the physical processes are known, it is possible (and desirable) to
+understand how different architecture classes are formed. As this
+paper is dedicated to planet formation and its imprints on archi-
+tecture, we briefly review the ingredients of the Bern model here.
+Readers interested in further details of this model are referred to
+the recent NGPPS series of papers (Emsenhuber et al. 2021a,b;
+Schlecker et al. 2021a; Burn et al. 2021; Schlecker et al. 2021b;
+Mishra et al. 2021). The historic development of the Bern model
+may be traced through the works of Alibert et al. (2004, 2005);
+Mordasini et al. (2009); Alibert et al. (2011); Mordasini et al.
+(2012a,b); Alibert et al. (2013); Fortier et al. (2013); Marboeuf
+et al. (2014b); Thiabaud et al. (2014); Dittkrist et al. (2014); Jin
+et al. (2014) and is reviewed in Benz et al. (2014); Mordasini
+(2018).
+Based on the core-accretion paradigm (Pollack et al. 1996),
+the Bern model is a global model of planet formation and evo-
+lution. The model studies the growth of several lunar-mass pro-
+toplanetary embryos embedded in protoplanetary disks (consist-
+ing of a gaseous and solid phase) around a solar-type star. The
+disk model is based on viscous angular momentum transport
+(Lynden-Bell & Pringle 1974; Veras & Armitage 2004; Hueso
+& Guillot 2005). Turbulence is characterised by the Shakura &
+Sunyaev (1973) approach. The initial mass of the solid disk de-
+pends on the metallicity of the star and also on the condensation
+state of the molecules in the disk (Thiabaud et al. 2014). The
+solids in the disk are composed of a swarm of rocky and icy
+planetesimals. The solids in the disk evolve via (a) accretion by
+growing planets, (b) interaction with gaseous disk, (c) dynamical
+stirring from planets and other planetesimals, and so on (Fortier
+et al. 2013). The 1D geometrically thin disk evolution is studied
+up to 1000 au.
+This star–disk–embryo numerical system is endowed with
+several physical processes, which are occurring simultaneously
+and in a self-consistently coupled way. Some of these physical
+processes are: stellar evolution (Baraffe et al. 2015), interactions
+between viscous protoplanetary disk and star (Lynden-Bell &
+Pringle 1974; Shakura & Sunyaev 1973; Clarke et al. 2001; Mat-
+suyama et al. 2003; Veras & Armitage 2004; Nakamoto & Nak-
+agawa 1994; Hueso & Guillot 2005), condensation of volatile
+and/or refractory species (Marboeuf et al. 2014b,a; Thiabaud
+et al. 2014), planet formation physics (Alibert et al. 2013; Fortier
+et al. 2013; Mordasini et al. 2012b), orbital and tidal migration
+(Coleman & Nelson 2014; Paardekooper et al. 2011; Dittkrist
+et al. 2014), gravitational N-body interactions (Chambers 1999;
+Alibert et al. 2013; Emsenhuber et al. 2021a,b), atmospheric es-
+cape (Jin et al. 2014), bloating (Sarkis et al. 2021), and so on
+(see Fig. 1 in Mishra et al. (2019) for a schematic diagram). In
+addition, the model also calculates the internal structure of all
+planets, assuming them all to be spherically symmetric.
+In the synthetic planetary population we use in the present
+work, some initial conditions are fixed, namely we use a 1M⊙
+mass star and a disk viscosity α = 2 × 10−3, describing the ini-
+tial shape of the gas and planetesimal disks via power laws (Ve-
+ras & Armitage 2004), with a planetesimal size of 300m, and a
+fixed density (rocky 3.2 g cm−3, icy 1 g cm−3). We add 100 pro-
+toplanetary embryos to the protoplanetary disk. We ensure that
+no two embryos start within 10 hill radii of each other (Kokubo
+& Ida 1998, 2002). This model is then run 1000 times while
+varying other initial conditions. We varied the initial gas mass in
+the protoplanetary disk, disk lifetime, stellar metallicity, disk in-
+ner edge, and the initial location of the protoplanetary embryos
+(for details see Emsenhuber et al. 2021b).
+The Bern model includes a significant variety of physics
+and uses plausible choices of initial conditions, which are mo-
+tivated by observations. However, it is only a simplified low-
+dimensional approximation of our current understanding of
+planet formation. For example, we model planet formation via
+core-accretion only and ignore other methods, such as disk insta-
+bility (Schib et al. 2021). Among others, we also assume that the
+dust-to-gas ratio is the same for both the host star and the disk,
+and that all dust in the disk is aggregated into planetesimals. The
+N-body interactions are tracked for only 20 Myr, which may be
+inadequate to capture dynamical effects occurring in the outer
+parts of the system. The assumptions, choices, and simplifica-
+tions made in this model may have a strong impact on the out-
+come of this paper. Nevertheless, exploring the implications of
+our architecture framework using synthetic populations via the
+Bern model is a necessary first step. The main result of this pa-
+per is not in understanding the formation of any single plane-
+tary system but to show that, for different architecture classes,
+discernible patterns of formation pathways emerge. Future stud-
+ies could apply our architecture framework (from Paper I) with
+other planet formation models. If the formation pathways for the
+Article number, page 3 of 12
+
+A&A proofs: manuscript no. 44705corr
+different architecture classes were found to remain the same af-
+ter using different formation models, then our results would be
+strengthened and become more robust.
+3. Nature: Role of star and disk initial conditions
+In this section, we study the connection between the initial con-
+ditions and the final architecture of a system. We begin by count-
+ing the number of different architecture classes that emerge from
+our population synthesis as a function of the various initial con-
+ditions that are varied. The role of varying disk masses and stel-
+lar metallicities is presented in Sect. 3.1, and that of varying disk
+lifetimes in Sect. 3.2. For completeness, we measure the relative
+count for an architecture class within a bin by dividing the num-
+ber of systems of a particular architecture class in a bin by the
+total number of systems in that bin. We emphasise that, as in Pa-
+per I, the architecture of a system is identified with its bulk mass
+architecture. Thus, when we refer to a similar or ordered sys-
+tem, we are referring to a system whose bulk mass architecture
+is similar or ordered, respectively.
+3.1. Protoplanetary disk: Mass and stellar metallicity
+Figure 2 (upper left) shows the dependence of the architecture
+class relative counts on the initial mass of gas in the protoplan-
+etary disk. Over 96% of all disks that started with gas masses
+≲ 0.04M⊙ give rise to planetary systems of similar architecture.
+About 1% of these low-mass disks lead to each of the other three
+architecture classes. The relative count of systems with similar
+architecture shows a clear decreasing trend with increasing mass
+in the disk gas.
+The production of the remaining three architecture classes
+tends to increase with increasing disk gas mass, but with dis-
+tinct trends. As the mass in the gas disk increases, the relative
+count of mixed architectures increases first, and then decreases
+for gas mass ≳ 0.12M⊙. The relative count for both anti-ordered
+and ordered architectures continues to increase with increasing
+disk mass. Anti-ordered architectures become the most common
+outcome from large disks with gas mass ≳ 0.12M⊙.
+In Fig. 2 (upper right), we see the binned relative count of
+different architecture classes as a function of the mass of the
+solids in the protoplanetary disk. This plot shows some of the
+same features that we saw in Fig. 2 (upper left). About 99% of all
+disks that have solid masses ≲ 200M⊕ give rise to similar plan-
+etary systems. The production of similar architecture decreases
+as the mass of solids in a disk is increased.
+Before continuing, we note that this is already a result of
+considerable importance. The physical processes encoded in the
+Bern model are the same for all 1000 planetary systems. The
+only difference between these synthetic systems arises from the
+variations in their initial conditions. We are seeing that almost
+all low-mass disks give rise to only one architecture, the similar
+class. This occurs despite all the physical processes that can act
+upon the system and induce some architectural variation. As we
+show below, the low mass of the disk limits some of the phys-
+ical processes that sculpt a system’s architecture. We conclude
+that the production of systems of the similar architecture class is
+dominated by initial conditions.
+Close to 60% of all observed systems in our multi-planetary
+systems catalogue (from Paper I) are similar in their mass ar-
+chitecture (Paper I). For some of these similar class systems
+(like Trappist-1, TOI-178, etc), if their formation is via core-
+accretion, our work may suggest strong limits on the initial mass
+of their protoplanetary disks.
+The relative count of the other three architecture classes in-
+creases as the solid mass in the disk increases. The production
+of mixed architectures peaks around disks of ≈ 1MJ and then
+decreases. The prevalence of anti-ordered and ordered architec-
+tures continues to increase with increasing disk mass. For heavy
+massive disks, anti-ordered architecture is the most common out-
+come.
+Figure 2 (middle left) shows the relative count of each ar-
+chitecture class in the synthetic population as a function of stel-
+lar metallicity. Figure 2 (middle right) shows the same for the
+41 observed multi-planetary systems. The selection criterion for
+our observed catalogue is detailed in Paper I. We find an inter-
+esting correlation between the metallicity and the architecture
+of a system, hereafter referred to as the metallicity–architecture
+correlation, and note the following trends. Over 98% of all sys-
+tems with Fe/H < −0.2 are of similar type. The relative count
+of similar architecture decreases as the metallicity is increased.
+The relative counts of the other three architecture classes are be-
+low 5% for metallicities ≤ −0.2. At different rates, the relative
+counts of mixed, ordered, and anti-ordered classes increase with
+increasing metallicity. Our catalogue of observed planetary sys-
+tems shows an encouragingly similar trend.
+Our observations catalogue suffers from detection biases and
+incompleteness. One way in which these limitations manifest
+is that we do not find any observed example of anti-ordered
+architecture. The qualitative trend for the relative count of ob-
+served system architectures as a function of their stellar metal-
+licity agrees with our synthetic systems. For example, the rela-
+tive count of similar observed systems decreases with increasing
+metallicity. The relative count of ordered architectures continues
+to increase with increasing metallicity.
+To understand the origin of these correlations, we study the
+relation between initial disk mass (both in solids and gases), stel-
+lar metallicity, and the final architecture of the systems in our
+model. In the Bern model, the initial solid mass of the disk is a
+fraction of the initial gas mass of the disk. This fraction is cor-
+related with the dust-to-gas ratio, which also depends on the gas
+mass itself because the location of different icelines depend on it.
+By simulating systems with varying dust-to-gas ratio (fD/G), we
+simulate systems around stars with different metallicities. This
+is due to the following relation:
+10[Fe/H] =
+fD/G
+fD/G,⊙
+,
+fD/G,⊙ = 0.0149 (Lodders 2003).
+(1)
+The metallicities in our simulations vary from −0.6 to 0.5 fol-
+lowing Santos et al. (2005).
+Figure 2 shows the solid disk mass as a function of the gas
+disk mass (bottom left) and the total mass in the planets as a
+function of the solid disk mass (bottom right). Each point rep-
+resents one planetary system, and the shape and colour of the
+marker shows its final architecture. These two plots help us un-
+derstand the correlations discussed above.
+The bottom left panel of Fig. 2 shows the relationship be-
+tween gas disk mass, solid disk mass, metallicity, and the final ar-
+chitecture of the system. Generally, when the mass of the solids
+in a disk is ≳ 1MJ(≈ 318M⊕), the production of architectures
+other than similar is triggered. We note that up to a certain gas
+disk mass (≲ 0.02M⊙), irrespective of the metallicity, all disks
+lead to similar architecture. For heavier gas disks (≳ 0.02M⊙),
+metallicities begin to play a role. If the gas disk mass is high
+enough, even low metallicities (≈ −0.2) can trigger the produc-
+tion of architectures other than the similar class. However, for
+lower gas disk masses, higher metallicities are required to pro-
+duce about a 1MJ mass in the solid disk.
+Article number, page 4 of 12
+
+L. Mishra et al.: Architecture Framework II – Nature versus nurture: Emergent formation pathways of architecture classes
+0.00
+0.04
+0.08
+0.12
+0.16
+Protoplanetary Disk: Gas Mass [M
+]
+0
+20
+40
+60
+80
+100
+Relative count of planetary systems [%]
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+0
+200
+400
+600
+800
+1000
+Protoplanetary Disk: Solid Mass [M
+]
+0
+20
+40
+60
+80
+100
+Relative count of planetary systems [%]
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+Metallicity [Fe/H]
+0
+20
+40
+60
+80
+100
+Relative count of planetary systems [%]
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+Metallicity [Fe/H]
+0
+20
+40
+60
+80
+100
+Relative count of planetary systems [%]
+Observations
+Similar
+Anti-Ordered
+Mixed
+Ordered
+10
+2
+10
+1
+Protoplanetary Disk: Gas Mass [M
+]
+10
+1
+10
+2
+10
+3
+Protoplanetary Disk: Solid Mass [M
+]
+1MJ
+318M
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+[Fe/H] = 0.5
+[Fe/H] = -0.6
+10
+1
+10
+2
+10
+3
+Protoplanetary Disk: Solid Mass [M
+]
+10
+0
+10
+1
+10
+2
+10
+3
+10
+4
+Total Mass in Planets [M
+]
+10 %
+100 %
+1MJ
+318M
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+Efficiency of
+solid accretion[%]
+Fig. 2. Role of disk mass and the metallicity–architecture correlation. The top two rows show the binned relative count of each architecture class
+as a function of initial disk gas mass (upper left), disk solid mass (upper right), stellar metallicity in the synthetic population (middle left), and
+stellar metallicity in observed systems (middle right). The length of the error bars corresponds to the total number of systems in each bin as:
+100/
+√
+bin counts. In the bottom panels, each point corresponds to a single planetary system. The system architecture is indicated by the colour
+and shape of the marker. The bottom left panel shows the solid mass in the disk as a function of the disk gas mass. The two diagonal lines convey
+the role of stellar metallicity. The dashed horizontal line indicates the mass of Jupiter. The bottom right panel shows the total mass in planets as a
+function of the solid mass in the protoplanetary disk. The two diagonal lines indicate the efficiency of converting solids from the disk into planets.
+If the planets in a hypothetical system could accrete all the solid mass of its disk, and these planets had no gaseous atmosphere, then such a system
+would lie on the diagonal line corresponding to 100% accretion efficiency. The dashed vertical line indicates the mass of Jupiter.
+Article number, page 5 of 12
+
+A&A proofs: manuscript no. 44705corr
+It is clear that the mass in the solids of the protoplanetary
+disk plays an essential role here. The bottom right panel of Fig.
+2 explains the above statement. The total mass in the planets
+increases as the mass of solids in the disk increases. When the
+mass of solids in the disk is ∼ 1MJ, the distribution of total mass
+in planets shows a jump. This is because massive planets can be-
+gin to accrete significant amounts of gas. For the core-accretion
+scenario, this plot suggests that similar architectures occur for
+low-mass disks because they cannot produce massive giant plan-
+ets. Gas giants are very effective in inducing dynamical stirring,
+which are in turn responsible for shaping the system architecture.
+This signifies the role played by physical processes in producing
+the mixed, anti-ordered, and ordered architectures1.
+3.2. Lifetime of the protoplanetary disk
+In this section, we explore the role of disk lifetime (i.e. the age
+of a protoplanetary disk) in defining the final architecture class
+of a system. The lifetime of a disk, in the Bern model, is influ-
+enced by the external disk photo-evaporation rate (see Emsenhu-
+ber et al. (2021a) for details) and the mass of the disk.
+Figure 3 (left) shows the binned relative count of system ar-
+chitecture as a function of disk lifetime. About 80% of all disks
+with lifetimes ranging from 1 to 5 Myr produce systems of the
+similar architecture class. The relative count of similar systems
+decreases as disk lifetime increases. The relative count of mixed
+architecture does not show any significant variation with disk
+lifetime. The relative counts of anti-ordered and ordered archi-
+tectures vary as the disk lifetime increases. This suggests that
+the physical mechanisms by which disks shape the final archi-
+tectures of systems play a role in shaping similar, anti-ordered,
+and ordered architectures.
+The trends of the relative counts of architecture classes with
+disk lifetime are similar to the distribution of relative counts as
+functions of disk mass. We would like to understand whether
+system architecture is influenced by disk lifetime directly or via
+an inherent dependence of disk lifetime on disk mass. The right
+panel of Fig. 3 shows the gas disk mass as a function of disk
+lifetime. The scatter plot depicting each individual disk shows
+that, generally, low-mass disks have short lifetimes. The solid
+lines depict the average gas mass for each architecture class for
+each disk lifetime bin.
+The gas mass of the disks that go on to form systems of
+mixed, anti-ordered, or ordered architecture shows a weak de-
+pendence on disk lifetime. On average, the more massive disks
+seem to last longer. For disks that give rise to the similar archi-
+tecture class, this trend is clearly visible. If more massive disks
+also live longer, this partly explains the relative count distribu-
+tion seen in Fig. 3 (left).
+However, disks also affect the planetary architecture in other
+interesting ways, namely orbital migration and eccentricity, and
+inclination damping. We study the effect of these planet–disk
+interactions in shaping system architecture in Sect. 4.1.
+1 The architecture framework is not sensitive to the absolute value of
+a planetary quantity, such as mass, but only the ratio of the quantities
+for adjacent planets. Independent of the architecture framework, we will
+present another system-level framework analysing the state of a plane-
+tary system. This other classification framework is sensitive to the abso-
+lute mass of a planet and will address the role of giant planets on system-
+level properties. The state classification framework reveals a drastic dif-
+ference between systems with and without giant planets (Mishra et al.
+in prep.).
+4. Nurture: Role of dynamical stirring
+Whether or not the final architecture of a planetary system is
+pre-determined by its initial conditions from the host star and
+the protoplanetary disk remains unclear. If not, the mechanism
+by which dynamical processes shape the architecture of a plane-
+tary system remains to be determined. It also remains unclear as
+to whether or not dynamical processes remove all traces of ini-
+tial conditions from the final system, or whether these stochastic
+processes leave their impressions on the final architecture. In this
+section, we try to answer these questions. We focus our attention
+on dynamical interactions between planets and the protoplane-
+tary disk, and the gravitational multi-body interactions amongst
+planets themselves.
+While there exist several dynamical mechanisms that shape
+the final architecture, we simplify the task before us by concen-
+trating on violent dynamical instabilities that change a planetary
+system in a non-trivial manner. For each synthetic planetary sys-
+tem, we count the number of planet–planet mergers, planetary
+ejections, and planets falling into their host star. We use these
+counts as a proxy to assess the strength of dynamical interactions
+that occur in a system. In the subsequent subsections, we study
+planet–disk interactions and planet–planet interactions (mergers,
+ejections, stellar accretion). These dynamical effects give rise to
+stochasticity and are thereby inherently unpredictable. However,
+we hope that the underlying dynamical processes that are sculpt-
+ing the system architecture emerge as patterns in the counts of
+these violent events.
+4.1. Planet–disk interactions
+Protoplanetary disks interact with planets via several mecha-
+nisms. Planets may experience orbital migration via gravitation
+interactions with the disk. Low-mass planets undergo type I mi-
+gration, which in the Bern model is implemented following the
+approaches of Coleman & Nelson (2014); Paardekooper et al.
+(2011). Massive planets may open a gap in the disk and undergo
+type II migration (Dittkrist et al. 2014). The disk also dampens
+the eccentricity and inclination of planets, which is coherently
+applied within the N-body integrator. Readers interested in the
+details of the implementation are referred to Emsenhuber et al.
+(2021a,b).
+Figure 4 (left) shows the count of mergers and ejections for
+each planetary system in our synthetic population as a function
+of the lifetime of its protoplanetary disk. For an easier visualisa-
+tion of any underlying trend, we also show the average merger
+and ejection counts for each disk lifetime bin. The number of
+planet–planet mergers shows a clear correlation with disk life-
+time. Disks that live longer usually give rise to planetary sys-
+tems that undergo more mergers than short-lived disks. We refer
+to this correlation as ‘migration assisted mergers’. One possible
+explanation for this correlation could be that disks allow plan-
+ets to migrate depending on their mass 2. Two adjacent planets
+that are not migrating at the same rate, perhaps owing to their
+different masses, can come close enough for a merger to occur.
+The number of ejections does not show any clear trend with disk
+lifetime. Disks dampen a planet’s eccentricity and inclination.
+As ejection requires extremely violent interactions (marked by
+2 There could be other scenarios which contribute to the ‘migration as-
+sisted mergers’ correlation. For example, migration may allow planets
+to become more massive by accreting more material due to increased
+access to planetesimals (Alibert et al. 2005). Massive planets may inter-
+act more amongst themselves, leading to more mergers.
+Article number, page 6 of 12
+
+L. Mishra et al.: Architecture Framework II – Nature versus nurture: Emergent formation pathways of architecture classes
+1.0
+1.8
+3.2
+5.6
+10.0
+17.8
+Protoplanetary Disk: Lifetime [Myr]
+0
+20
+40
+60
+80
+100
+Relative count of planetary systems [%]
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+1.0
+1.8
+3.2
+5.6
+10.0
+17.8
+Disk Lifetime [Myr]
+10
+2
+10
+1
+Protoplanetary Disk: Gas Mass [M
+]
+Bern Model
+Similar
+Anti-Ordered
+Mixed
+Ordered
+Fig. 3. Role of disk lifetime on system architecture. Left: Binned relative counts of architecture classes as a function of disk lifetime. The length of
+error bars corresponds to the total number of systems in each bin, as: 100/
+√
+bin counts. Right: Scatter plot shows the disk gas mass as a function
+of disk lifetime. The solid lines show the binned average gas disk mass for each architecture class.
+1.0
+1.6
+2.6
+4.2
+6.8
+11.0
+17.8
+Disk Lifetime [Myr]
+0
+20
+40
+60
+80
+100
+Counts
+Bern Model
+Mergers
+Ejections
+0
+20
+40
+60
+80
+100
+Planet counts
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Density
+Bern Model
+Ejections: w/ planet-disk interactions
+ Mergers: w/ planet-disk interactions
+Ejections: w/o planet-disk interactions
+ Mergers: w/o planet-disk interactions
+Fig. 4. Effect of planet–disk interactions on architecture. Left: Scatter plot shows the number of planet–planet mergers and planetary ejections that
+occurred in systems as a function of disk lifetime. The solid lines show the average counts for each disk lifetime bin. Right: Distribution of the
+total number of mergers (dashed) and ejections (solid) for the entire synthetic population. The black line depicts the nominal synthetic population,
+and the red line depicts a different synthetic population in which the disk-)planet interactions were artificially switched off.
+high eccentricities and inclinations), disks may essentially in-
+hibit planetary ejections.
+To test these ideas, we simulated another population of 1000
+planetary systems. In this population (NG140), planet–disk in-
+teractions (gas-driven migrations, and eccentricity and inclina-
+tion damping) are artificially switched off. For all such systems,
+we count the number of mergers and ejections and compare them
+with our nominal population. Figure 4 (right) shows the distribu-
+tion of the number of planet–planet mergers and planetary ejec-
+tions in the two populations.
+As expected, the number of planet–planet mergers decreases
+(distribution shifts to the left) when planet–disk interactions are
+switched off. This confirms the migration-assisted mergers cor-
+relation presented above. The distribution of ejections, on the
+other hand, increases significantly when planet–disk interactions
+are switched off. When the damping of the planetary eccentricity
+and inclination by the disk is switched off, the gravitational in-
+teractions between planets increases, such that many planets are
+ejected.
+We make two observations from the results presented so
+far. First, counts of mergers and ejections seem to be a good
+proxy for the prevalence of dynamical interactions, as they cap-
+ture some of the well-established dynamical effects concern-
+ing planet–disk interactions. Second, we observe that disks af-
+Article number, page 7 of 12
+
+A&A proofs: manuscript no. 44705corr
+fect system architecture in a multitude of ways. While disk
+mass shows a direct relation to final architecture, disks also af-
+fect system architecture indirectly by influencing the dynami-
+cal interactions that occur therein. Long-living disks give rise to
+more mergers and inhibit planetary ejections. Conversely, sys-
+tems emerging from short-lived disks experience fewer mergers.
+4.2. Planet–planet interactions
+Above, we show that planet–disk interactions in the Bern model
+may influence the dynamical interactions occurring in a system.
+Now, in this section, we are interested in understanding how
+these violent events shape the final architecture of a system.
+Planets interact with each other gravitationally. These multi-
+body interactions are tracked via a N-body integrator in the Bern
+model. The end result of some of the more violent interactions
+is that planets are lost via one of several channels: planet–planet
+mergers3, planetary ejections, accretion by the host star, and so
+on. These channels allow a planetary system to fundamentally
+alter itself and its architecture.
+Figure 5 shows, for each architecture class, the distribution
+of planet–planet mergers and the number of planets lost via ejec-
+tions and stellar accretion. At first glance, losing planets to the
+host star may not seem appropriate for planet–planet interac-
+tions. However, many of these planets meet their fate, in the
+Bern model, when they are pushed inwards after being captured
+in mean-motion resonances with other planets4. Therefore, this
+channel of losing planets is included here. We caution the reader
+that the absolute number of planets lost via any channel is model-
+dependent. The quantity of interest here is the relative difference
+between the different architecture classes.
+Figure 5 suggests that the similar architecture class is almost
+completely shaped by planet–planet mergers. Most similar sys-
+tems in our simulations have between 40 and 80 mergers taking
+place within them, and the median number of mergers is 63. Vi-
+olent dynamical interactions that lead to the ejection of planets
+seems to be very rare in this architecture type, as 100% of all
+similar systems lose less than five planets via planetary ejection
+(median ejections is 0). Likewise, similar systems seem to not
+rely on the stellar accretion channel for losing planets (median
+stellar accretions is 0).
+Systems with mixed architecture also undergo many planet–
+planet mergers. The number of mergers in mixed systems ranges
+from 50 to 85, and the median of mergers is 70. In a clear con-
+trast from similar architectures, the ejection and stellar accretion
+channels play an important role for mixed systems. The median
+number of planets lost via ejections is 7, and via stellar accre-
+tions is 2.
+anti-ordered systems utilise all three dynamical channels.
+The distribution of mergers in anti-ordered systems is roughly
+similar to that of mixed systems. The range is between 50 and 85
+and the median number of mergers is 67. However, anti-ordered
+systems tend to lose more planets via the ejection channel. The
+number of planets lost via dynamical ejection ranges from 0 to
+35 with a median value of 14.5. Compared to mixed systems,
+3 In our model, when the distance between two planets becomes
+smaller than the sum of their radii, a planet–planet collision is said to oc-
+cur. We treat such merger events in a simplified manner: the cores of the
+target–impactor pair are merged, the lesser massive body loses its enve-
+lope, and the impact energy is added to the merged new body following
+Broeg & Benz (2012), which determines what part of the gaseous enve-
+lope is ejected.
+4 The model also includes inward migration of planets as a result of
+the stellar tides.
+anti-ordered systems also tend to lose more planets via stellar
+accretion (median is 6).
+Amongst the four architecture classes, ordered systems seem
+to undergo the greatest number of dynamical interactions. The
+distribution of planet–planet mergers in ordered systems shows
+a tail-like feature. The number of mergers ranges from 55 to 85,
+with 62 being the median. All ordered systems eject at least five
+planets. The number of ejections has a range from 5 to 35, and
+the median is 23. The distribution of planets lost via the stel-
+lar accretion channel shows a shift to the right. The number of
+planets accreted by the star ranges from 0 to 20 with 8 being the
+median.
+A comprehensive picture of the role of dynamical history in
+shaping the final architecture emerges from the four panels in
+Fig. 5. Similar systems tend to rely only on the merger channel
+for shaping their system architecture. As planetary systems in
+all four architecture classes undergo a considerable number of
+mergers, this channel may not suffice to explain or distinguish
+the emergence of the four architecture classes. This is in line
+with what was found before, namely that the emergence of the
+similar class is mostly governed by the initial conditions.
+While initial conditions seem to decide whether a system be-
+comes similar or one of the other three architectures, there ap-
+pears to be a trend in the role of dynamical interactions in shap-
+ing mixed, anti-ordered, and ordered architectures. The distri-
+butions of the ejection and accretion channels distinguish these
+three architectures. These distributions show a shift to the right,
+indicating that more planets are being lost via these two channels
+as we move from mixed to anti-ordered and to ordered architec-
+tures. Thus, we conclude that if initial conditions do not allow
+a system to become similar, its fate is decided by its dynamical
+history, among other effects. If the strength of the dynamical in-
+teractions increases in a system, the architecture of the system
+changes from mixed to anti-ordered or to ordered.
+All systems in the Bern model start with 100 protoplane-
+tary embryos. Above, we show that systems of different archi-
+tecture show varying propensity to lose planets via the different
+dynamical channels. This suggests that we should also see an ef-
+fect of the dynamical history of the four architecture classes in
+their multiplicity distribution. We observed this effect in Fig. 6
+of Paper I. We do not have a way to determine the initial num-
+ber of embryos of the planetary systems we observe today. Our
+approach may therefore not be directly applicable to observed
+planetary systems. We remind the reader that while the quanti-
+tative aspects we present in this section are probably model de-
+pendent, the qualitative nature of these results is of paramount
+importance.
+5. The Aryabhata formation scenario
+In this section, we propose a planet-formation scenario to ex-
+plain a feature observed by Paper I (Sect. 5.4). We found that
+many synthetic planetary systems have a peculiar water-mass-
+fraction architecture namely that all planets hosted in these sys-
+tems are water-rich worlds. We explain this peculiar feature with
+the ‘Aryabhata formation scenario’.
+The first exoplanets to be discovered were hot Jupiters —
+giant planets orbiting their host stars at very short periods
+(Mayor & Queloz 1995). Orbital migration was suggested as
+a possible mechanism to explain these short periods (Lin et al.
+1996; Lin & Ida 1997). Theoretical studies indicate that orbital
+migration and planet–star tidal interactions should make many
+close-in planets unstable. In the 1990s, Doug Lin described ‘the
+last of the Mohicans’ scenario (Garaud 2011). In this scenario,
+Article number, page 8 of 12
+
+L. Mishra et al.: Architecture Framework II – Nature versus nurture: Emergent formation pathways of architecture classes
+0
+20
+40
+60
+80
+100
+Planet Counts
+0
+20
+40
+60
+80
+100
+Distribution of systems [%]
+Similar Systems
+Merged
+Ejected
+Star Accreted
+0
+20
+40
+60
+80
+100
+Planet Counts
+0
+20
+40
+60
+80
+100
+Distribution of systems [%]
+Mixed Systems
+Merged
+Ejected
+Star Accreted
+0
+20
+40
+60
+80
+100
+Planet Counts
+0
+20
+40
+60
+80
+100
+Distribution of systems [%]
+Anti-Ordered Systems
+Merged
+Ejected
+Star Accreted
+0
+20
+40
+60
+80
+100
+Planet Counts
+0
+20
+40
+60
+80
+100
+Distribution of systems [%]
+Ordered Systems
+Merged
+Ejected
+Star Accreted
+Fig. 5. Effect of planet–planet interactions on system architecture. For each architecture class, the panels show a histogram of the counts of planet–
+planet mergers, ejections, and stellar accretion occurring in the synthetic population. The y-axis in all panels is scaled to reflect the percentage of
+systems in each of the four architecture classes. For example, 100% of all similar systems lost less than five planets via planetary ejection.
+the protoplanetary disk gives rise to planets, many of which are
+doomed to fall onto the star. The surviving observable planets
+are those that were able to escape annihilation.
+For some simulated systems, we noticed a modified version
+of this scenario. Protoplanetary disks seem to give rise to planets
+at different epochs. In the first epoch, several intermediate-mass
+planets (1 − 100M⊕) are formed within the first 1Myr. Most of
+these ‘first generation’ planets are subsequently lost mainly via
+giant impacts (and a few are lost via orbital or tidal migration
+leading to stellar accretion). This purging phase is catastrophic
+to all planets that started within the ice line. Over the next few
+million years, a second epoch sees the advent of a ‘second gen-
+eration’ of planets. Most of these second-generation planets are
+born outside the ice line, and are able to migrate inwards dur-
+ing the disk lifetime. After disk dissipation, migration comes to
+a halt and many of these planets survive long-term N-body evo-
+lution in our simulations. We call this the Aryabhata formation
+scenario. The key difference between the two scenarios is that
+in the Aryabhata formation scenario (a) planets (surviving and
+lost) are born in different epochs, and (b) most first-generation
+planets are lost via giant impacts.
+We quantify this scenario with the Aryabhata’s number, µ,
+which is the ratio of the surviving planets that started inside the
+ice line to the total number of surviving planets:
+Aryabhata’s number: µ =
+n(astart
+embryo ≤ aice)
+n
+.
+(2)
+At the start of our calculations, all systems have an Aryabhata’s
+number ≈ 0.5 ± 0.1. Figure 12 of Paper I (middle) shows the ice
+mass fraction architecture of simulated planetary systems. The
+colour of each point shows the Aryabhata’s number.
+Most planetary systems with CS ( fice) ≈ CV( fice) ≈ 0 have
+µ close to zero. This suggests that most (or all) of the surviv-
+ing planets in such systems started outside the ice line. The for-
+mation path of these systems falls into the Aryabhata formation
+scenario. These classes of systems can be identified by two char-
+acteristics: (i) the core water-mass fraction for different planets
+in these systems is similar, and (ii) the core water-mass fraction
+for most planets is high (owing to their origin outside the ice
+line) making them water-rich planets. Approximately, one-fifth
+of the simulated systems fall into this scenario. Among these,
+about half are of similar class, one-third are anti-ordered, and
+the remaining systems have either a mixed or ordered mass ar-
+chitecture.
+There exists an almost linear relationship between CV( fice)
+and µ. Using scipy’s linear regression module, we obtain a slope
+of 1.8 and intercept of 0.18 between these two quantities. The
+correlation coefficient is R = 0.95, indicating a strong correlation
+between the Aryabhata’s number and the coefficient of variation
+of core water mass fraction. This suggests a possibility to iden-
+tify observed exoplanetary systems that may have originated via
+the Aryabhata formation scenario. By determining the CV( fice)
+of a system, the Aryabhata’s number can be estimated. Systems
+with low µ values probably arose from this scenario.
+For systems that fall into the default scenario (positive
+CS ( fice), implying an increasing core water mass fraction inside-
+out), the Aryabhata’s number is µ > 0. We note that most sys-
+tems with µ ⪆ 0.6 show similarity in their mass architecture.
+Overall, the intra-system core water-mass-fraction architec-
+ture of most planetary systems seems to take one of two forms.
+(i) Those characterised by CS ( fice) ≈ CV( fice) ≈ 0 and µ = 0.
+These systems are composed of water-rich planets wherein the
+core water mass fraction is similar across the different planets.
+All surviving planets in these systems started outside the ice line.
+The Aryabhata formation scenario explains these systems. (ii)
+Those with CS ( fice) > 0 and µ > 0. These systems represent
+the ‘default’ or common outcome of our simulations. The plan-
+etary core water-mass fraction in these systems increases from
+one planet to another with increasing distance from the host star.
+Some of the surviving planets started from inside the ice line.
+At the extreme end, systems in which 60% or more surviving
+planets started inside the ice line tend to have a similar mass
+architecture.
+6. Summary, conclusions, and future work
+Paper I of this series introduced a novel, model-independent
+framework for characterising the architecture of planetary sys-
+tems at the system level. Planetary-system architectures can be
+separated into four classes: similar, mixed, anti-ordered, and or-
+dered. This classification is achieved via two quantities: the co-
+efficient of similarity and the coefficient of variation. The math-
+ematical CS versus CV architecture space was found to have
+forbidden regions – regions in which no planetary system can
+exist. In Paper I, the mass architecture classes of observed and
+synthetic systems were characterised. The mass architecture of
+synthetic systems was compared with their radii architecture,
+bulk-density architecture, core-mass architecture, spacing archi-
+tecture, and water-mass-fraction architecture. As in Paper I, we
+identify a system’s architecture with its mass architecture.
+In this paper, we explore the core-accretion-based formation
+pathways —around a solar-like star— of the four classes of plan-
+etary system architecture. We tried to disentangle the role of
+nature (initial conditions of planet formation) from that of nur-
+Article number, page 9 of 12
+
+A&A proofs: manuscript no. 44705corr
+ture (physical processes occurring during planet formation). Our
+findings can be summarised as follows:
+1. System-level analysis: Our findings show that a system-
+level analysis of planetary system architecture via our ar-
+chitecture framework (Paper I) provides an abundance of in-
+formation. We show that planetary formation and evolution
+process leave their imprint on the entire system architecture.
+2. Solid disk mass: The initial amount of solids in the proto-
+planetary disk in our models plays an important role in decid-
+ing the architectural fate of a planetary system. Disks with a
+solid mass (initial content of planetesimals) of ≲ 1MJ almost
+always give rise to systems with similar architecture. Mixed
+architectures arise most often from disks with solid masses
+≈ 1MJ. Disks with solid mass ≳ 1MJ favour the production
+of anti-ordered and ordered architectures.
+3. Gas disk mass and metallicity: Initial gas disk mass and
+stellar metallicity influences the final architecture of a plane-
+tary system by controlling the initial mass of solids in the
+disk. Metallicity, in our models, is simply related to the
+dust-to-gas ratio, which allows us to convert a fraction of
+the initial gas disk mass into initial dust mass (eq. 1). Ap-
+plying the architecture framework on the synthetic systems
+from the Bern model allows us to predict the existence of
+a metallicity–architecture correlation. The observed correla-
+tion between metallicity and final architecture is in qualita-
+tive agreement with the Bern model.
+4. Metallicity–architecture correlation: The architecture of a
+planetary system correlates with the metallicity of the host
+star. Most systems hosted by a low-metallicity star (Fe/H <
+−0.2) are of similar architecture. As the metallicity of the
+star increases, mixed, ordered, and anti-ordered architectures
+become increasingly common.
+5. Disk lifetime: The occurrence of systems of a similar ar-
+chitecture around short-lived disks is high, and their fre-
+quency reduces around long-lived disks. The frequency of
+anti-ordered architecture increases as disk lifetime increases.
+These correlations are mediated in at least two ways. First,
+disks interact with planets, where orbital migration and ec-
+centricity and inclination damping occur. Due to the ‘mi-
+gration assisted merger’ correlation, long-lasting disks allow
+planetary systems to have, in general, more planet–planet
+mergers and inhibit planetary ejections. These dynamical
+events shape a system’s final architecture. In addition, in our
+model, disk lifetimes are correlated with disk masses, which
+also strongly influences the system architecture.
+6. Dynamical interactions: Planetary systems can signifi-
+cantly alter their architecture via (at least) three dynamical
+channels: planet–planet mergers, planetary ejections, and ac-
+cretion by the host star. All architecture classes in our forma-
+tion model were found to undergo numerous merger events.
+Similar systems rely entirely on mergers to shape their final
+architecture. As the strength of the dynamical interactions
+experienced by a system (quantified by the number of ejec-
+tions and/or accretions) increases, the architecture of a sys-
+tem shifts from mixed to anti-ordered to ordered.
+7. The Aryabhata formation scenario: Systems following
+this formation scenario have the following formation path-
+way. First-generation planets (formed within 1 Myr) are lost
+mostly via giant impacts. Second-generation planets started
+outside the ice line and migrated inwards. The surviving
+planets are from the second generation and shape the ar-
+chitecture of the system. This scenario explains about 20%
+of simulated systems in which the core water-mass-fraction
+architecture is different from the default scenario. Systems
+following this formation scenario (i) host only those planets
+that have a high core water-mass fraction and (ii) host only
+those planets that started outside the ice line. We introduce
+the Aryabhata’s number to identify those systems that follow
+this formation scenario and find that 80% of all anti-ordered
+simulated systems are formed via the Aryabhata formation
+scenario.
+8. Nature versus nurture: Overall, our model suggests that
+initial conditions —or ‘nature’— dictate whether a system
+will have a similar architecture or one of the other three ar-
+chitecture classes, namely mixed, anti-ordered, or ordered
+(via initial disk mass). If nature does not allow a system
+to have a similar mass architecture, then the final architec-
+ture is controlled by ‘nurture’, or dynamical interactions,
+among other possible effects. As the dynamical interactions
+increase, the final architecture tends to become mixed, anti-
+ordered, and then ordered.
+We would like to offer readers some warning when interpret-
+ing our results. Although the architecture framework (from Pa-
+per I) is model-independent, the present results hinge critically
+on the underlying planet formation model – the Bern model.
+There are several assumptions, simplifications, and choices to
+be made when simulating synthetic planetary systems using the
+Bern model. For example, the treatment of planet–planet merg-
+ing collisions is relatively simple (Ali-Dib et al. 2022). We also
+assume simplified planet-formation conditions; that is, our star–
+disk–planet system is isolated enough so that we may ignore the
+influence of the stellar neighbourhood, stellar flybys, and so on
+(Bate 2012, 2018). The main strength of this study does not lie in
+providing an explanation of the formation pathway of any partic-
+ular system. Instead, our main result is the observation that when
+groups of planetary systems are identified (architecture classes),
+general trends in formation pathways emerge. This allowed us to
+map the roles of nature and nurture in shaping the final architec-
+ture of a planetary system.
+The results of this study can be strengthened or challenged
+in several observational and theoretical ways. We list some pos-
+sibilities for future studies emerging from this work:
+1. Linking disk mass distribution and architecture occur-
+rence rates: Our model suggests that there should be a direct
+relationship between the mass of the solid disk and the final
+architecture of a system. While initial disk masses and the
+final architecture of the same system will forever remain un-
+observable, this relation can be tested statistically. The dis-
+tribution of initial disk masses and the distribution of final
+system architecture can be linked by formation models. We
+speculate that in future, when these two distributions become
+available, formation models can be used to predict one or the
+other. In fact, this problem can also be turned around; we can
+identify the right family of models as those that correctly link
+the observed distributions of protoplanetary disk masses and
+architecture occurrence rates. We believe such tests are cru-
+cial for the development and eventual emergence of a stan-
+dard model for exoplanetary astrophysics.
+2. Metallicity–architecture correlation: Our work suggests
+that the current architecture of a planetary system should be
+related to the metallicity of its host star. As both of these
+are observable, testing this metallicity–architecture correla-
+tion should be feasible. Here, we used a catalogue of 41 ob-
+served multi-planet systems (from Paper I) to test this corre-
+lation. We find a qualitative agreement between theory and
+Article number, page 10 of 12
+
+L. Mishra et al.: Architecture Framework II – Nature versus nurture: Emergent formation pathways of architecture classes
+observations. However, our observational catalogue suffers
+from incompleteness and low-number statistics, which pre-
+vents us from making any further assertions. More obser-
+vational data are required to confirm or reject the proposed
+metallicity-architecture correlation. It would also be interest-
+ing to estimate the current architecture occurrence rate based
+on the known metallicity distributions.
+3. Confirming formation pathways: Confirming the forma-
+tion pathways discovered in the present study with obser-
+vations is challenging. However, the strength of our results
+will increase if different planet-formation models are stud-
+ied through the architecture framework. Hence, one possible
+line of future work involves repeating the present study using
+different planet-formation models.
+4. Extending the architecture framework: So far, we have
+calibrated our classification scheme for the mass architec-
+tures only. Calibrating the architecture classification frame-
+work on other quantities maybe useful. Especially for plan-
+etary radii, which are observable via transit surveys, the use
+of machine learning methods may be necessary.
+5. Temporal evolution of system architecture: In the nomi-
+nal Bern model population studied in this paper, protoplane-
+tary embryos of 100 lunar masses are initialised in the pro-
+toplanetary disk at the start. This necessarily implies that all
+planetary systems start as similar type systems. It would be
+interesting to inquire whether this is generally true in nature
+as well. If this is the case, this implies that the ‘default’ ar-
+chitecture of all planetary systems is similar and the phys-
+ical processes playing out in the system evolve this archi-
+tecture into other possibilities. Investigating this may lead to
+deep insights into the structure of planetary system architec-
+ture. In addition, such studies would be necessary to interpret
+the observed architecture occurrences, as observed planetary
+systems are seldom of the same age.
+6. External perturbations: Stellar flybys or multi-planetary
+systems around binaries provide excellent theoretical and ob-
+servational laboratories with which to study the influence of
+external perturbations on the architecture of planetary sys-
+tems. This problem, when turned around, is also useful in
+deducing the perturbed or dynamical (or lack of) history of
+observed planetary systems.
+This paper presents new insights obtained by analysing plan-
+etary systems at the system-level. We showed that several pat-
+terns emerged in the formation pathways of the four architecture
+classes. These patterns linked the initial conditions of planet for-
+mation with the final architecture of a system – bridging the vast
+temporal gap of several billions of years between the birth of
+planets to their final assembly.
+Acknowledgements. This work has been carried out within the frame of the Na-
+tional Centre for Competence in Research PlanetS supported by the Swiss Na-
+tional Science Foundation. We acknowledge the support of the Swiss National
+Fund under grant 200020_172746 and 200021_204847 “PlanetsInTime”. LM ac-
+knowledges the generous hospitality of the "Planet Formation" workshop by the
+Munich Institute for Astro-, Particle and BioPhysics (MIAPbP) which is funded
+by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
+under Germany’s Excellence Strategy – EXC-2094 – 390783311.
+Data: The synthetic planetary populations (NGPPS) used in this work are avail-
+able online at http://dace.unige.ch. Software: Python (Van Rossum &
+Drake 2009), NumPy (Oliphant 2006), Seaborn (Waskom & the seaborn de-
+velopment team 2020), Pandas (pandas development team 2020), Matplotlib
+(Hunter 2007).
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+arXiv:2301.00030v1  [math-ph]  28 Dec 2022
+Duality family of KdV equation
+Xin Gu,a Yuan-Yuan Liu,b Wen-Du Li,c,1 and Wu-Sheng Daia,2
+aDepartment of Physics, Tianjin University, Tianjin 300350, P.R. China
+bTheoretical Physics Division, Chern Institute of Mathematics, Nankai University, PR China
+cCollege of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, PR China
+Abstract: It is revealed that there exist duality families of the KdV type equation. The
+duality family consists of an infinite number of the generalized KdV (GKdV) equation.
+A duality transformation relates the GKdV equations in a duality family. Once a family
+member is solved, the duality transformation presents the solutions of all other family
+members. We show some dualities as examples, such as the soliton solution-soliton solution
+duality and the periodic solution-soliton solution duality.
+1liwendu@tjnu.edu.cn
+2daiwusheng@tju.edu.cn
+
+Contents
+1
+Introduction
+1
+2
+Duality family of GKdV equation
+3
+3
+Duality family of KdV equation: Example
+5
+4
+Conclusion
+7
+1
+Introduction
+After Russell found the solitary wave phenomenon, and studying nonlinear evolution equa-
+tions began in physics and mathematics [1]. When Kortoweg and de Vries studied the water
+wave in the long-wave approximation and finite small amplitude, they gave the Korteweg-de
+Vries (KdV) equation [1–3],
+∂u
+∂t − 6u∂u
+∂x + ∂3u
+∂x3 = 0.
+(1.1)
+The KdV equation is a basic model in nonlinear evolution equations [4, 5].
+The KdV
+equation defines many physical phenomena, such as waves in anharmonic crystals [6], waves
+in bubble liquid mixtures [7], ion acoustic waves [8–10], and waves in warm plasma [8–10].
+Soliton solution. The solitary wave solutions of the KdV equation are noted as solitons.
+The velocity of the solitary wave relates to its magnitude [11], and after the collision, it re-
+tains the original magnitude, shape, and velocity [12, 13]. The theory of solitons emerges in
+biochemistry, nonlinear optics, mathematical biosciences, fluid dynamics, plasma physics,
+nuclear physics, and geophysics [14]. There have been many approaches to calculating the
+soliton solution [15, 16], such as the Painlevé analysis method, Bäcklund transformation
+method, Hirota bilinear method, inverse scattering method, and Darboux transformation
+method [1]. These methods apply not only to calculating the soliton solution of the KdV
+equation but also to other partial differential equations [17]. These methods have differ-
+ent limits in applications, and there is no universal method for solving nonlinear partial
+differential equations generally [18].
+Modified KdV (mKdV) equation and generalized KdV (GKdV) equation.
+The KdV
+equation is a special case of the GKdV equation. The GKdV equation generally is [19]
+∂u
+∂t − f (u) ∂u
+∂x + ∂3u
+∂x3 = 0.
+(1.2)
+The GKdV equation recovers the KdV equation (1.1) when f (u) = 6u.
+A special GKdV equation with f (u) = −αuk is the KdV type equation with a power-
+law nonlinearity [20],
+∂u
+∂t + αuk ∂u
+∂x + ∂3u
+∂x3 = 0,
+(1.3)
+– 1 –
+
+and the mKdV equation is Eq. (1.3) with k = 2 and α = 6 [21]. The Miura transforma-
+tion establishes a one-to-one correspondence between the solutions of the KdV equation
+and the solutions of the mKdV equation [22]. The mKdV equation has a rich physical
+background [23, 24]. The mKdV equation can describe a bounded particle propagating in
+a one-dimensional nonlinear lattice with a harmonic force [25], small amplitude ion acous-
+tic waves propagating in plasma physics [8], and the thermal pulse propagating through a
+single crystal of sodium fluoride [26, 27].
+Duality and duality family. Newton in Principia revealed a duality between gravitation
+and elasticity in classical mechanics, now called the Newton-Hooke duality [28]. E. Kasner
+and V.I. Arnol’d independently find the generalized duality between power potentials: two
+power potentials U (r) = ξra and V (r) = ηrA are dual if a+2
+2
+=
+2
+A+2, called the Kasner-
+Arnol’d theorem [29–31].
+Recently, we find that such a duality generally exists in classical mechanics, quantum
+mechanics, and scalar fields and present the duality among arbitrary potentials [32]. We
+find that the duality is not a duality only between two potentials but exists duality families
+[32]. Each duality family consists of an infinite number of potentials dual to each other.
+Each duality family consists of an infinite number of potentials; in a duality family, every
+potential is dual to all other potentials. Once a family member’s solution is obtained, we can
+obtain all other members’ solutions by the duality transformation. Therefore, the duality
+relation can be used to find the solutions for classical mechanics, quantum mechanics, field
+theory, and nonlinear equations (such as the Gross-Pitaevskii equation) [33–35]. The duality
+can also be used to classify long-range potentials in quantum mechanics [36].
+In this paper, we reveal duality and duality families for the GKdV equation.
+The
+duality transformation can transform the solution of a GKdV equation into the solution of
+its dual GKdV equation. The GKdV equation duality family consists of an infinite number
+of GKdV equations that are dual to each other. The solution of all GKdV equations in a
+duality family can be obtained from the solution of one solved family member by the duality
+transformation. This way, we can obtain a series of exact solutions of GKdV equations.
+As an example, we discuss the KdV equation duality family in which the KdV equation
+is a member. As an example, we discuss the KdV equation duality family in which the
+KdV equation (1.1) and the KdV type equation with a power-law nonlinearity (1.3) are
+family members. The duality transformation gives a series of 1-soliton solutions of GKdV
+equations from a 1-soliton solution of the KdV equation (1.1). We also consider the duality
+between the periodic solution of the KdV equation and the soliton solution of the mKdV
+equation.
+In particular, since the solution of all GKdV equations in a duality family can be
+obtained from the solution of one family member by the duality transformation, we can
+develop an indirect approach for solving GKdV equations: (1) constructing the duality
+family of this equation; (2) looking for an ‘easy’ equation in the duality family and solving
+the ‘easy’ equation; (3) solving the wanted equation by the duality transformation.
+In section 2, we present the duality and duality family of the GKdV equation. In section
+3, we consider two examples: (1) solving the KdV equation with a power-law nonlinearity
+from the KdV equation by the duality transformation; (2) the duality between the periodic
+– 2 –
+
+solution of the KdV equation and the soliton solution of the mKdV equation. The conclusion
+is given in section 4.
+2
+Duality family of GKdV equation
+In this section, we give the duality and duality family of the traveling wave GKdV equation.
+The solutions of a GKdV equation can be obtained from its dual equation by the duality
+transformation.
+The traveling wave with a velocity C of the GKdV equation (1.2) is given by
+d3u
+dz3 + [C − f (u)] du
+dz = 0.
+(2.1)
+where u (x, t) = u (z) and z = x + Ct.
+The traveling wave GKdV equation (2.1) has the following duality relation.
+Two traveling wave GKdV equations,
+d3u
+dz3 + [C − f (u)] du
+dz = 0,
+(2.2)
+d3v
+dζ3 + [C − g (v)] dv
+dζ = 0,
+(2.3)
+if
+1
+C u−2 [G − U (u) − Fu] = 1
+C v−2 [G − V (v) − Fv] ,
+(2.4)
+where
+d2U (u)
+du2
+= −f (u) ,
+(2.5)
+d2V (v)
+dv2
+= −g (v) ,
+(2.6)
+F = −
+�d2u
+dz2 + Cu + dU (u)
+du
+�
+,
+(2.7)
+F = −
+�d2v
+dζ2 + Cv + dV (v)
+dv
+�
+,
+(2.8)
+G = 1
+2
+�du
+dz
+�2
++ 1
+2Cu2 + U (u) + Fu,
+(2.9)
+G = 1
+2
+�dv
+dζ
+�2
++ 1
+2Cv2 + V (v) + Fv,
+(2.10)
+then their solutions satisfy
+u ↔ vσ,
+(2.11)
+z ↔
+�
+C
+C σζ.
+(2.12)
+– 3 –
+
+Here σ is an arbitrarily chosen constant.
+Integral of motion. Before going on, we first illustrate the meaning of G, F, G, and F,
+taking G and F as examples.
+Broadly speaking, G and F are both integrals of motion for the equation of motion (2.2).
+In principle, the integral of the equation of motion over time is known as the integral of
+motion. Here G and F are integration constants of integrating the traveling wave equation
+(2.2) over z and u, respectively; we here still call them integral of motion.
+Multiplying both sides of the GKdV equation (2.2) by dz and integrating, and using
+(2.5) give d2u
+dz2 + Cu + dU(u)
+du
+= −F, i.e., Eq. (2.7), where F is the integration constant of
+the integral over z.
+Similarly, multiplying both sides of (2.7) by du and integrating give 1
+2
+� du
+dz
+�2 + 1
+2Cu2 +
+U (u) + Fu = G, i.e., (2.9), where G is the integration constant of the integral over u and
+�
+du d2u
+dz2 =
+�
+dz du
+dz
+d2u
+dz2 = 1
+2
+�
+dz d
+dz
+�du
+dz
+�2 = 1
+2
+� du
+dz
+�2 is used.
+Proof of duality relation. Substituting the duality transformations (2.11) and (2.12)
+into (2.7) gives
+C
+C
+d2v
+dζ2 + C
+C (σ − 1) v−1
+�dv
+dζ
+�2
++ σCv + v2(1−σ) dU (vσ)
+dv
++ σv1−σF = 0.
+(2.13)
+By (2.9), we have
+C
+C (σ − 1) v−1
+�dv
+dζ
+�2
+= 2 (σ − 1) v1−2σ [G − U (vσ) − Fvσ] − C (σ − 1) v.
+(2.14)
+Using (2.14) to eliminate the term (σ − 1) v−1 �
+dv
+dζ
+�2
+in (2.13), we arrive at
+C
+C
+d2v
+dζ2 + Cv + 2 (σ − 1) v1−2σ [G − U (vσ) − Fvσ] + v2(1−σ) dU (vσ)
+dv
++ σv1−σF = 0. (2.15)
+By the duality transformation (2.4), we can obtain
+V (v) = G − Fv − C
+C v2−2σ [G − U (vσ) − Fvσ] .
+(2.16)
+Taking the derivative of (2.16) with respect to v gives
+dV (v)
+dv
+= −F + 2 C
+C (σ − 1) v1−2σ [G − U (vσ) − Fvσ] + C
+C v2(1−σ)
+�dU (vσ)
+dv
++ σvσ−1F
+�
+.
+(2.17)
+Substituting (2.17) into (2.15) gives
+d2v
+dζ2 + Cv + dV (v)
+dv
++ F = 0.
+(2.18)
+Then taking the derivative with respect to ζ and using (2.6), we arrive at (2.3).
+Discussion of U. The relation between f (u) in the GKdV equation (2.2) and U (u) in
+(2.5) is not unique. U (u; a, b) = U (u) + au + b and U (u) lead to the same f (u), and both
+correspond to the GKdV equation (1.2).
+– 4 –
+
+The integral of motion F, corresponding to U (u; a, b), by (2.7), is F (a, b) = −
+�
+d2u
+dz2 + Cu + dU(u;a,b)
+du
+�
+=
+F − a; the integral of motion G, corresponding to U (u; a, b) , by (2.9), is G (a, b) =
+1
+2
+�du
+dz
+�2 + 1
+2Cu2 + U (u; a, b) + F (a, b) u = G + b. Therefore, by (2.4), the duality transfor-
+mation given by U (u; a, b) is
+1
+C u−2 [G (a, b) − U (u; a, b) − F (a, b) u] = 1
+C v−2 [G − V (v; a, b) − Fv] .
+(2.19)
+Here V (v; a, b) is the duality of U (u; a, b).
+Substituting U (u; a, b), F (a, b), and G (a, b) into the duality transformation (2.19)
+gives
+V (v; a, b) = G − Fv − C
+C v2−2σ [G − U (vσ) − Fvσ] = V (v) .
+(2.20)
+That is, in the GKdV equation, although the correspondence between f (u) and U (u) is
+not unique, the same f (u) corresponding to different U (u), the choice of U (u) does not
+influence the duality of the GKdV equation.
+3
+Duality family of KdV equation: Example
+We consider a special duality family of the GKdV equation as an example in this section.
+The KdV equation and mKdV equation are family members of this duality family. The
+solutions of all family members in a duality family are related by a duality transformation.
+In a duality family containing the KdV equation, we can solve all the GKdV equations
+in the family from the solution of the KdV equation by the duality transformation. In
+this section, we give the solution of the KdV equation with a power-law nonlinearity from
+the solution of the KdV equation; the mKdV equation is the power-law nonlinearity KdV
+equation with power 2.
+Duality family of the KdV equation and the KdV equation with a power-law nonlinearity.
+The KdV equation (1.1) with z = x − Ct,
+d3u
+dz3 − (C + 6u) du
+dz = 0,
+(3.1)
+has a 1-soliton solution [37]
+u (z) = −C
+2 sech2
+�√
+C
+2 z
+�
+.
+(3.2)
+The soliton solution is a localized traveling wave solution. The localization solution, taking
+the 1-soliton solution as an example, means that (3.2) when z → ±∞, u (z) → 0. The
+integral of motion of the 1-soliton solution (3.2), by (2.7), (2.9) and (3.2), is
+F = 0 and G = 0.
+(3.3)
+Then the dual equation of the traveling wave KdV equation given by the duality transfor-
+mation (2.4) is
+d3v
+dζ3 −
+�
+C + C
+C (2 + σ) (1 + σ) vσ
+� dv
+dζ = 0.
+(3.4)
+– 5 –
+
+Since σ can be chosen arbitrarily, (3.4) is not a single equation but forms a duality family.
+All the GKdV equations labeled by different σ in the duality family are dual equations of
+the KdV equation.
+By (2.11) and (2.12), we can obtain the solution of Eq. (3.4)
+v (ζ) =
+�
+−C
+2 sech2
+�√
+C
+2 σζ
+��1/σ
+,
+(3.5)
+where ζ = x − Ct has a velocity −C.
+Instead of z, rewrite the dual equation (3.4) by (t, x):
+∂v
+∂t + αvσ ∂v
+∂x + ∂3v
+∂x3 = 0,
+(3.6)
+where α = − C
+C (2 + σ) (1 + σ). When σ is taken as a positive integer, (3.6) is the KdV
+equation with a power-law nonlinearity, and the solution (3.5) becomes
+v (x, t) =
+�
+−C
+2 sech2
+�√
+C
+2 σ (x − Ct)
+��1/σ
+,
+(3.7)
+or equivalently, v (x, t) =
+�
+C(2+σ)(1+σ)
+2α cosh2� √
+C
+2 σ(x−Ct)
+�
+�1/σ
+, which agrees with Ref. [38].
+In this duality family, the family member σ = 1 is the KdV equation (1.1), and the
+family member σ = 2 is the mKdV equation
+∂v
+∂t − 12 C
+C v2 ∂v
+∂x + ∂3v
+∂x3 = 0.
+(3.8)
+(3.7) with σ = 2 gives the 1-soliton solution of the mKdV equation (3.8)
+v (x, t) = ±
+�
+−C
+2 sech
+�√
+C (x − Ct)
+�
+.
+(3.9)
+Now, by the duality relation, we have obtained all family members’ solutions from the KdV
+equation’s solution.
+Periodic solution-soliton solution duality. A duality exists between the periodic solution
+and the soliton solution of the GKdV equation. We take the periodic solution of the KdV
+equation and the soliton solution of the mKdV equation as an example.
+The KdV equation (1.1) has a periodic solution
+u (x, t) = 1
+6C
+�
+1 + 3 tan2
+�√
+C
+2
+(x − Ct)
+��
+.
+(3.10)
+The KdV equation (1.1) with z = x − Ct becomes (3.1), and its solution (3.10) becomes
+u (z) = C
+6
+�
+1 + 3 tan2
+�C
+2 z
+��
+(3.11)
+– 6 –
+
+with the period
+2π
+√
+C .
+The integral of motion of the periodic solution (3.10) of the KdV equation, by (2.7),
+(2.9) and (3.10), is
+F = 0,
+G = −C3
+54 .
+(3.12)
+The dual equation of the traveling wave KdV equation given by the duality transformation
+(2.4) is then
+d3v
+dζ3 +
+�
+C − 1
+27 (1 − σ) (1 − 2σ) CC2v−2σ + C
+C (σ + 1) (σ + 2) vσ
+� dv
+dζ = 0,
+(3.13)
+where ζ = x+Ct. The duality transformations (2.11) and (2.12) give the solution of (3.13).
+v (ζ) =
+�
+C
+6
+�
+1 − 3 tanh2
+�√
+C
+2 σζ
+���1/σ
+.
+(3.14)
+σ running over all possible values gives all equations and their solutions in the duality
+family.
+The family member σ = 1 and C = −C in the duality family is the KdV equation
+(1.1). Different from the 1-soliton solution (3.4), however, the family member σ = −1 is
+the traveling wave mKdV equation
+d3v
+dζ3 + C
+�
+1 − 2
+9C2v2
+� dv
+dζ = 0.
+(3.15)
+or, with ζ = x + Ct and C = 27
+C2 ,
+∂v
+∂t − 6v2 ∂v
+∂x + ∂3v
+∂x3 = 0,
+(3.16)
+which, by (3.14), has a traveling wave solution
+v (x, t) =
+2
+√
+C
+√
+3
+�
+1 − 3 tanh2 � √
+C
+2 (x + Ct)
+��.
+(3.17)
+It can be directly verified that v (x, t) → −
+√
+3C
+3
+when x, t → ±∞, so (3.17) is a soliton
+solution of the mKdV equation (3.16).
+In this example, the duality of the periodic solution is a soliton solution.
+Indirect approach for solving equations. The existence of the duality family gives us
+an indirect approach to solving equations. When solving an equation, we can (1) find its
+duality family; (2) look for and solve an ‘easy’ family member, and (3) achieve the solution
+of this equation by the duality transformation.
+4
+Conclusion
+This paper reveals a duality among the GKdV equations, and all the GKdV equations that
+are dual to each other form a duality family. In a duality family, the solutions of different
+family members are related by the duality transformation.
+– 7 –
+
+In a duality family, we only need to solve one family member, and the duality trans-
+formation can give solutions for all other family members. This allows us to develop an
+indirect approach to solving the GKdV equation.
+In this paper, as an example, we discuss the GKdV equation duality family containing
+the KdV equation and the KdV equation with a power-law nonlinearity: seeking 1-soliton
+solution of the KdV equation with a power-law nonlinearity from a 1-soliton solution of
+the KdV equation by the duality relation. In another example, we consider the periodic
+solution-soliton solution duality. By the duality transformation, we give a soliton solution
+of the mKdV equation from a periodic solution of the KdV equation.
+Acknowledgments
+We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supported
+in part by Special Funds for theoretical physics Research Program of the NSFC under Grant
+No. 11947124, and NSFC under Grant Nos. 11575125 and 11675119.
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+10
+

2dAyT4oBgHgl3EQfPvZl/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf,len=470
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='00030v1  [math-ph]  28 Dec 2022  Duality family of KdV equation  Xin Gu,a Yuan-Yuan Liu,b Wen-Du Li,c,1 and Wu-Sheng Daia,2  aDepartment of Physics, Tianjin University, Tianjin 300350, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' China  bTheoretical Physics Division, Chern Institute of Mathematics, Nankai University, PR China  cCollege of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, PR China  Abstract: It is revealed that there exist duality families of the KdV type equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The  duality family consists of an infinite number of the generalized KdV (GKdV) equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  A duality transformation relates the GKdV equations in a duality family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Once a family  member is solved, the duality transformation presents the solutions of all other family  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' We show some dualities as examples, such as the soliton solution-soliton solution  duality and the periodic solution-soliton solution duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  1liwendu@tjnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='cn  2daiwusheng@tju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='cn  Contents  1  Introduction  1  2  Duality family of GKdV equation  3  3  Duality family of KdV equation: Example  5  4  Conclusion  7  1  Introduction  After Russell found the solitary wave phenomenon, and studying nonlinear evolution equa-  tions began in physics and mathematics [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' When Kortoweg and de Vries studied the water  wave in the long-wave approximation and finite small amplitude, they gave the Korteweg-de  Vries (KdV) equation [1–3],  ∂u  ∂t − 6u∂u  ∂x + ∂3u  ∂x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1)  The KdV equation is a basic model in nonlinear evolution equations [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The KdV  equation defines many physical phenomena, such as waves in anharmonic crystals [6], waves  in bubble liquid mixtures [7], ion acoustic waves [8–10], and waves in warm plasma [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Soliton solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The solitary wave solutions of the KdV equation are noted as solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The velocity of the solitary wave relates to its magnitude [11], and after the collision, it re-  tains the original magnitude, shape, and velocity [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The theory of solitons emerges in  biochemistry, nonlinear optics, mathematical biosciences, fluid dynamics, plasma physics,  nuclear physics, and geophysics [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' There have been many approaches to calculating the  soliton solution [15, 16], such as the Painlevé analysis method, Bäcklund transformation  method, Hirota bilinear method, inverse scattering method, and Darboux transformation  method [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' These methods apply not only to calculating the soliton solution of the KdV  equation but also to other partial differential equations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' These methods have differ-  ent limits in applications, and there is no universal method for solving nonlinear partial  differential equations generally [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Modified KdV (mKdV) equation and generalized KdV (GKdV) equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The KdV  equation is a special case of the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The GKdV equation generally is [19]  ∂u  ∂t − f (u) ∂u  ∂x + ∂3u  ∂x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2)  The GKdV equation recovers the KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) when f (u) = 6u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  A special GKdV equation with f (u) = −αuk is the KdV type equation with a power-  law nonlinearity [20],  ∂u  ∂t + αuk ∂u  ∂x + ∂3u  ∂x3 = 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3)  – 1 –  and the mKdV equation is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3) with k = 2 and α = 6 [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The Miura transforma-  tion establishes a one-to-one correspondence between the solutions of the KdV equation  and the solutions of the mKdV equation [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The mKdV equation has a rich physical  background [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The mKdV equation can describe a bounded particle propagating in  a one-dimensional nonlinear lattice with a harmonic force [25], small amplitude ion acous-  tic waves propagating in plasma physics [8], and the thermal pulse propagating through a  single crystal of sodium fluoride [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Duality and duality family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Newton in Principia revealed a duality between gravitation  and elasticity in classical mechanics, now called the Newton-Hooke duality [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Kasner  and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Arnol’d independently find the generalized duality between power potentials: two  power potentials U (r) = ξra and V (r) = ηrA are dual if a+2  2  =  2  A+2, called the Kasner-  Arnol’d theorem [29–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Recently, we find that such a duality generally exists in classical mechanics, quantum  mechanics, and scalar fields and present the duality among arbitrary potentials [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' We  find that the duality is not a duality only between two potentials but exists duality families  [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Each duality family consists of an infinite number of potentials dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Each duality family consists of an infinite number of potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' in a duality family, every  potential is dual to all other potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Once a family member’s solution is obtained, we can  obtain all other members’ solutions by the duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Therefore, the duality  relation can be used to find the solutions for classical mechanics, quantum mechanics, field  theory, and nonlinear equations (such as the Gross-Pitaevskii equation) [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The duality  can also be used to classify long-range potentials in quantum mechanics [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In this paper, we reveal duality and duality families for the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The  duality transformation can transform the solution of a GKdV equation into the solution of  its dual GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The GKdV equation duality family consists of an infinite number  of GKdV equations that are dual to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The solution of all GKdV equations in a  duality family can be obtained from the solution of one solved family member by the duality  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' This way, we can obtain a series of exact solutions of GKdV equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  As an example, we discuss the KdV equation duality family in which the KdV equation  is a member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' As an example, we discuss the KdV equation duality family in which the  KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) and the KdV type equation with a power-law nonlinearity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3) are  family members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The duality transformation gives a series of 1-soliton solutions of GKdV  equations from a 1-soliton solution of the KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' We also consider the duality  between the periodic solution of the KdV equation and the soliton solution of the mKdV  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In particular, since the solution of all GKdV equations in a duality family can be  obtained from the solution of one family member by the duality transformation, we can  develop an indirect approach for solving GKdV equations: (1) constructing the duality  family of this equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (2) looking for an ‘easy’ equation in the duality family and solving  the ‘easy’ equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (3) solving the wanted equation by the duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In section 2, we present the duality and duality family of the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' In section  3, we consider two examples: (1) solving the KdV equation with a power-law nonlinearity  from the KdV equation by the duality transformation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (2) the duality between the periodic  – 2 –  solution of the KdV equation and the soliton solution of the mKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The conclusion  is given in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  2  Duality family of GKdV equation  In this section, we give the duality and duality family of the traveling wave GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The solutions of a GKdV equation can be obtained from its dual equation by the duality  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The traveling wave with a velocity C of the GKdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2) is given by  d3u  dz3 + [C − f (u)] du  dz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1)  where u (x, t) = u (z) and z = x + Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The traveling wave GKdV equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) has the following duality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Two traveling wave GKdV equations,  d3u  dz3 + [C − f (u)] du  dz = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2)  d3v  dζ3 + [C − g (v)] dv  dζ = 0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3)  if  1  C u−2 [G − U (u) − Fu] = 1  C v−2 [G − V (v) − Fv] ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4)  where  d2U (u)  du2  = −f (u) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='5)  d2V (v)  dv2  = −g (v) ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='6)  F = −  �d2u  dz2 + Cu + dU (u)  du  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7)  F = −  �d2v  dζ2 + Cv + dV (v)  dv  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='8)  G = 1  2  �du  dz  �2  + 1  2Cu2 + U (u) + Fu,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9)  G = 1  2  �dv  dζ  �2  + 1  2Cv2 + V (v) + Fv,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='10)  then their solutions satisfy  u ↔ vσ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='11)  z ↔  �  C  C σζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='12)  – 3 –  Here σ is an arbitrarily chosen constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Integral of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Before going on, we first illustrate the meaning of G, F, G, and F,  taking G and F as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Broadly speaking, G and F are both integrals of motion for the equation of motion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In principle, the integral of the equation of motion over time is known as the integral of  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Here G and F are integration constants of integrating the traveling wave equation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2) over z and u, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' we here still call them integral of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Multiplying both sides of the GKdV equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2) by dz and integrating, and using  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='5) give d2u  dz2 + Cu + dU(u)  du  = −F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=', Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7), where F is the integration constant of  the integral over z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Similarly, multiplying both sides of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7) by du and integrating give 1  2  � du  dz  �2 + 1  2Cu2 +  U (u) + Fu = G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=', (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9), where G is the integration constant of the integral over u and  �  du d2u  dz2 =  �  dz du  dz  d2u  dz2 = 1  2  �  dz d  dz  �du  dz  �2 = 1  2  � du  dz  �2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Proof of duality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Substituting the duality transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='12)  into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7) gives  C  C  d2v  dζ2 + C  C (σ − 1) v−1  �dv  dζ  �2  + σCv + v2(1−σ) dU (vσ)  dv  + σv1−σF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='13)  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9), we have  C  C (σ − 1) v−1  �dv  dζ  �2  = 2 (σ − 1) v1−2σ [G − U (vσ) − Fvσ] − C (σ − 1) v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='14)  Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='14) to eliminate the term (σ − 1) v−1 �  dv  dζ  �2  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='13), we arrive at  C  C  d2v  dζ2 + Cv + 2 (σ − 1) v1−2σ [G − U (vσ) − Fvσ] + v2(1−σ) dU (vσ)  dv  + σv1−σF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='15)  By the duality transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4), we can obtain  V (v) = G − Fv − C  C v2−2σ [G − U (vσ) − Fvσ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='16)  Taking the derivative of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='16) with respect to v gives  dV (v)  dv  = −F + 2 C  C (σ − 1) v1−2σ [G − U (vσ) − Fvσ] + C  C v2(1−σ)  �dU (vσ)  dv  + σvσ−1F  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='17)  Substituting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='17) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='15) gives  d2v  dζ2 + Cv + dV (v)  dv  + F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='18)  Then taking the derivative with respect to ζ and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='6), we arrive at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Discussion of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The relation between f (u) in the GKdV equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2) and U (u) in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='5) is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) = U (u) + au + b and U (u) lead to the same f (u), and both  correspond to the GKdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  – 4 –  The integral of motion F, corresponding to U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b), by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7), is F (a, b) = −  �  d2u  dz2 + Cu + dU(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='a,b)  du  �  =  F − a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' the integral of motion G, corresponding to U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) , by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9), is G (a, b) =  1  2  �du  dz  �2 + 1  2Cu2 + U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) + F (a, b) u = G + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4), the duality transfor-  mation given by U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) is  1  C u−2 [G (a, b) − U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) − F (a, b) u] = 1  C v−2 [G − V (v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) − Fv] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='19)  Here V (v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) is the duality of U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Substituting U (u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b), F (a, b), and G (a, b) into the duality transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='19)  gives  V (v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' a, b) = G − Fv − C  C v2−2σ [G − U (vσ) − Fvσ] = V (v) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='20)  That is, in the GKdV equation, although the correspondence between f (u) and U (u) is  not unique, the same f (u) corresponding to different U (u), the choice of U (u) does not  influence the duality of the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  3  Duality family of KdV equation: Example  We consider a special duality family of the GKdV equation as an example in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The KdV equation and mKdV equation are family members of this duality family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The  solutions of all family members in a duality family are related by a duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In a duality family containing the KdV equation, we can solve all the GKdV equations  in the family from the solution of the KdV equation by the duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' In  this section, we give the solution of the KdV equation with a power-law nonlinearity from  the solution of the KdV equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' the mKdV equation is the power-law nonlinearity KdV  equation with power 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Duality family of the KdV equation and the KdV equation with a power-law nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) with z = x − Ct,  d3u  dz3 − (C + 6u) du  dz = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1)  has a 1-soliton solution [37]  u (z) = −C  2 sech2  �√  C  2 z  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2)  The soliton solution is a localized traveling wave solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The localization solution, taking  the 1-soliton solution as an example, means that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2) when z → ±∞, u (z) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The  integral of motion of the 1-soliton solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2), by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='2), is  F = 0 and G = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='3)  Then the dual equation of the traveling wave KdV equation given by the duality transfor-  mation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4) is  d3v  dζ3 −  �  C + C  C (2 + σ) (1 + σ) vσ  � dv  dζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4)  – 5 –  Since σ can be chosen arbitrarily, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4) is not a single equation but forms a duality family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  All the GKdV equations labeled by different σ in the duality family are dual equations of  the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='12), we can obtain the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4)  v (ζ) =  �  −C  2 sech2  �√  C  2 σζ  ��1/σ  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='5)  where ζ = x − Ct has a velocity −C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Instead of z, rewrite the dual equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4) by (t, x):  ∂v  ∂t + αvσ ∂v  ∂x + ∂3v  ∂x3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='6)  where α = − C  C (2 + σ) (1 + σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' When σ is taken as a positive integer, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='6) is the KdV  equation with a power-law nonlinearity, and the solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='5) becomes  v (x, t) =  �  −C  2 sech2  �√  C  2 σ (x − Ct)  ��1/σ  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7)  or equivalently, v (x, t) =  �  C(2+σ)(1+σ)  2α cosh2� √  C  2 σ(x−Ct)  �  �1/σ  , which agrees with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In this duality family, the family member σ = 1 is the KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1), and the  family member σ = 2 is the mKdV equation  ∂v  ∂t − 12 C  C v2 ∂v  ∂x + ∂3v  ∂x3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='8)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7) with σ = 2 gives the 1-soliton solution of the mKdV equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='8)  v (x, t) = ±  �  −C  2 sech  �√  C (x − Ct)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9)  Now, by the duality relation, we have obtained all family members’ solutions from the KdV  equation’s solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Periodic solution-soliton solution duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' A duality exists between the periodic solution  and the soliton solution of the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' We take the periodic solution of the KdV  equation and the soliton solution of the mKdV equation as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) has a periodic solution  u (x, t) = 1  6C  �  1 + 3 tan2  �√  C  2  (x − Ct)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='10)  The KdV equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1) with z = x − Ct becomes (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1), and its solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='10) becomes  u (z) = C  6  �  1 + 3 tan2  �C  2 z  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='11)  – 6 –  with the period  2π  √  C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The integral of motion of the periodic solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='10) of the KdV equation, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='7),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='10), is  F = 0,  G = −C3  54 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='12)  The dual equation of the traveling wave KdV equation given by the duality transformation  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4) is then  d3v  dζ3 +  �  C − 1  27 (1 − σ) (1 − 2σ) CC2v−2σ + C  C (σ + 1) (σ + 2) vσ  � dv  dζ = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='13)  where ζ = x+Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The duality transformations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='12) give the solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  v (ζ) =  �  C  6  �  1 − 3 tanh2  �√  C  2 σζ  ���1/σ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='14)  σ running over all possible values gives all equations and their solutions in the duality  family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  The family member σ = 1 and C = −C in the duality family is the KdV equation  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Different from the 1-soliton solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='4), however, the family member σ = −1 is  the traveling wave mKdV equation  d3v  dζ3 + C  �  1 − 2  9C2v2  � dv  dζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='15)  or, with ζ = x + Ct and C = 27  C2 ,  ∂v  ∂t − 6v2 ∂v  ∂x + ∂3v  ∂x3 = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='16)  which, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='14), has a traveling wave solution  v (x, t) =  2  √  C  √  3  �  1 − 3 tanh2 � √  C  2 (x + Ct)  ��.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='17)  It can be directly verified that v (x, t) → −  √  3C  3  when x, t → ±∞, so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='17) is a soliton  solution of the mKdV equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In this example, the duality of the periodic solution is a soliton solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Indirect approach for solving equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' The existence of the duality family gives us  an indirect approach to solving equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' When solving an equation, we can (1) find its  duality family;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' (2) look for and solve an ‘easy’ family member, and (3) achieve the solution  of this equation by the duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  4  Conclusion  This paper reveals a duality among the GKdV equations, and all the GKdV equations that  are dual to each other form a duality family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' In a duality family, the solutions of different  family members are related by the duality transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  – 7 –  In a duality family, we only need to solve one family member, and the duality trans-  formation can give solutions for all other family members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' This allows us to develop an  indirect approach to solving the GKdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  In this paper, as an example, we discuss the GKdV equation duality family containing  the KdV equation and the KdV equation with a power-law nonlinearity: seeking 1-soliton  solution of the KdV equation with a power-law nonlinearity from a 1-soliton solution of  the KdV equation by the duality relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' In another example, we consider the periodic  solution-soliton solution duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' By the duality transformation, we give a soliton solution  of the mKdV equation from a periodic solution of the KdV equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  Acknowledgments  We are very indebted to Dr G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Zeitrauman for his encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' This work is supported  in part by Special Funds for theoretical physics Research Program of the NSFC under Grant  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' 11947124, and NSFC under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' 11575125 and 11675119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content='  References  [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
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+page_content=' Ablowitz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Clarkson, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
+page_content=' Clarkson, Solitons, nonlinear evolution  equations and inverse scattering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2dAyT4oBgHgl3EQfPvZl/content/2301.00030v1.pdf'}
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+arXiv:2301.01925v1  [math.NT]  5 Jan 2023
+SELBERG’S CENTRAL LIMIT THEOREM OF L-FUNCTIONS NEAR
+THE CRITICAL LINE
+YOONBOK LEE
+Abstract. We find an asymptotic expansion of a multi-dimensional version of Sel-
+berg’s central limit theorem for L-functions on σ = 1
+2 + (log T )−θ and t ∈ [T, 2T ],
+where 0 < θ < 1
+2 is a constant.
+1. Introduction
+Selberg’s central limit theorem says that the function
+log ζ(σ + it)
+�
+π �
+p<t p−2σ
+has a Gaussian distribution in the complex plane for 1
+2 ≤ σ ≤ σT(θ), where
+σT := σT (θ) := 1
+2 +
+1
+(log T)θ
+for θ > 0 throughout the paper. See [8, Theorem 6.1] for a proof and [6] for a simple
+proof for the real part. It also holds for other L-functions. See [7, Theorem 2] for a
+general statement.
+When σ = σT and T ≤ t ≤ 2T, we have more precise estimations for the distribution
+of log ζ(σ + it) in [2] and [5] as follows.
+Theorem 1.1. [5, Theorem 1.2 and Lemma 2.3] Let 0 < θ < 1
+2, a < b and c < d be
+real numbers. There exist constants ǫ, κ > 0 and a sequence {dk,ℓ}k,ℓ≥0 of real numbers
+such that
+(1.1)
+1
+T meas{t ∈ [T, 2T] : log ζ(σT + it)
+√πψT
+∈ [a, b] × [c, d]}
+=
+�
+k+ℓ≤ǫψT
+dk,ℓ
+√ψT
+k+ℓ
+� b
+a
+e−πu2Hk(√πu)du
+� d
+c
+e−πv2Hℓ(√πv)dv + O
+�
+1
+(log T)κ
+�
+as T → ∞, where meas denotes the Lebesgue measure on R,
+ψT :=
+�
+p
+�
+k≥1
+1
+k2p2kσT
+Date: January 6, 2023.
+2010 Mathematics Subject Classification. 11M41.
+Key words and phrases. Central limit theorem, joint distribution of L-functions.
+1
+
+2
+YOONBOK LEE
+and Hn(x) is the n-th Hermite polynomial defined by
+(1.2)
+Hn(x) := (−1)nex2 dn
+dxn(e−x2).
+Moreover, d0,0 = 1, dk,ℓ = 0 for k + ℓ = 1, 2 and dk,ℓ = O(δ−k−ℓ
+0
+) for some δ0 > 0 and
+all k, ℓ.
+The leading term of the expansion in (1.1) is
+� b
+a
+e−πu2du
+� d
+c
+e−πv2dv,
+which is Gaussian, and the lower order terms may be evaluated using
+� b
+a
+e−πu2Hk(√πu)du = −1
+√π
+�
+e−πb2Hk−1(√πb) − e−πa2Hk−1(√πa)
+�
+for k ≥ 1. Note that the sequence {dk,ℓ} is defined by the generating series (2.19) in
+[5] and ψT = θ log log T + O(1) by the prime number theorem. It might be interesting
+to compare the asymptotic expansion in (1.1) with an Edgeworth expansion in the
+probability theory. See [1, Chapter 7] for more information.
+In this paper, we generalize Theorem 1.1 to a multi-variate setting for the L-
+functions L1, . . . , LJ satisfying the following assumptions:
+A1: (Euler product) For j = 1, . . . , J and Re(s) > 1 we have
+Lj(s) =
+�
+p
+d
+�
+i=1
+�
+1 − αj,i(p)
+ps
+�−1
+,
+where |αj,i(p)| ≤ pη for some fixed 0 ≤ η < 1
+2 and for every i = 1, . . . , d.
+A2: (Functional equation) The functions L1, L2, . . . , LJ satisfy the same functional
+equation
+Λj(s) = ωΛj(1 − ¯s),
+where
+Λj(s) := Lj(s)Qs
+k
+�
+ℓ=1
+Γ(λℓs + µℓ),
+|ω| = 1, Q > 0, λℓ > 0 and µℓ ∈ C with Re(µℓ) ≥ 0.
+A3: (Ramanujan hypothesis on average)
+�
+p≤x
+d
+�
+i=1
+|αj,i(p)|2 = O(x1+ǫ)
+holds for every ǫ > 0 and for every j = 1, . . . , J as x → ∞.
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+3
+A4: (Zero density hypothesis) Let Nf(σ, T) be the number of zeros of f(s) in Re(s) ≥
+σ and 0 ≤ Im(s) ≤ T. Then there exist positive constants κ1, κ2 such that for
+every j = 1, . . . , J and all σ ≥ 1
+2 we have
+NLj(σ, T) ≪ T 1−κ1(σ− 1
+2 )(log T)κ2.
+A5: (Selberg orthogonality conjecture) By assumption A1 we can write
+log Lj(s) =
+�
+p
+∞
+�
+k=1
+βLj(pk)
+pks
+.
+Then for all 1 ≤ j, k ≤ J, there exist constants ξj > 0 and cj,k such that
+�
+p≤x
+βLj(p)βLk(p)
+p
+= δj,kξj log log x + cj,k + O
+�
+1
+log x
+�
+,
+where δj,k = 0 if j ̸= k and δj,k = 1 if j = k.
+The assumptions A1–A5 are standard and expected to hold for all L-functions arising
+from automorphic representation for GL(n). In particular, they are verified by GL(1)
+and GL(2) L-functions, which are the Riemann zeta function, Dirichlet L-functions,
+L-functions attached to Hecke holomorphic or Maass cusp forms. Assumption A4 is
+weaker than the Riemann hypothesis, but it is strong enough to find a short Dirichlet
+approximation to each log Lj(σT + it) for almost all t ∈ [T, 2T]. For example, see [4,
+Lemma 4.2] for a proof. Assumption A5 insures the statistical independence of the
+log Lj(σT + it) for j = 1, . . . , J.
+Assuming assumptions A1–A5 for L1, . . . , LJ, we want to find an asymptotic ex-
+pansion for
+(1.3)
+1
+T meas{t ∈ [T, 2T] : log Lj(σT + it)
+�
+πψj,T
+∈ [aj, bj] × [cj, dj] for all j = 1, . . . , J},
+where
+(1.4)
+ψj,T := ξjθ log log T
+with the constants ξj in assumption A5 and aj, bj, cj, dj are real numbers for all j =
+1, . . . , J. Let
+L(s) :=
+�
+log |L1(s)|, . . . , log |LJ(s)|, arg L1(s), . . . , arg LJ(s)
+�
+and
+RT :=
+J�
+j=1
+[aj
+�
+πψj,T, bj
+�
+πψj,T] ×
+J�
+j=1
+[cj
+�
+πψj,T, dj
+�
+πψj,T],
+then (1.3) equals to
+ΦT(RT ) := 1
+T meas{t ∈ [T, 2T] : L(σT + it) ∈ RT }.
+
+4
+YOONBOK LEE
+Theorem 1.2. Let 0 < θ < 1
+2. Assume assumptions A1–A5 for L1, . . . , LJ. Then there
+exist constants ǫ, κ > 0 and a sequence {bk,l} of real numbers such that
+(1.5)
+ΦT(RT) =
+�
+K(k+l)≤ǫ log log T
+bk,l
+J�
+j=1
+1
+�
+ψj,T
+kj+ℓj
+×
+J
+�
+j=1
+� � bj
+aj
+e−πu2Hkj(√πu)du
+� dj
+cj
+e−πv2Hℓj(√πv)dv
+�
++ O
+�
+1
+(log T)κ
+�
+,
+where k = (k1, . . . , kJ) and l = (ℓ1, . . . , ℓJ) are vectors in (Z≥0)J and K(k) := k1 +
+· · · + kJ. Moreover, b0,0 = 1, bk,l = 0 if K(k + l) = 1 and bk+l = O(δ−K(k+l)
+0
+) for some
+δ0 > 0 and all k, l.
+Theorem 1.2 will be proved in the beginning of Section 2. Theorem 1.2 is essentially
+the same as Theorem 2.1 in [3], but it looks that the expansion in Theorem 1.2 is
+longer. Moreover, since the paper [3] contains only a sketched proof, our proof should
+be useful.
+Unlike dk,ℓ in Theorem 1.1, bk,l in Theorem 1.2 may not be zero for K(k + l) = 2.
+One reason is that ψT in Theorem 1.1 and ψj,T in Theorem 1.2 are different up to
+a constant order, even though they are asymptotically same. Moreover, when J > 1,
+there are additional terms essentially from the constants cj,k in assumption A5.
+Since the leading term in (1.5) is Gaussian and the other nonvanishing terms are
+O
+�
+1
+log log T
+�
+, we obtain the following corollary.
+Corollary 1.3. Let 0 < θ < 1
+2. Assume assumptions A1–A5 for L1, . . . , LJ. Then we
+have
+ΦT(RT ) =
+J�
+j=1
+� � bj
+aj
+e−πu2du
+� dj
+cj
+e−πv2dv
+�
++ O
+�
+1
+log log T
+�
+.
+We will prove theorems and propositions in Section 2 and lemmas in Section 3. We
+conclude the introduction with a summary of notations:
+• σT = σT (θ) = 1
+2 +
+1
+(log T)θ and 0 < θ < 1
+2.
+• k = (k1, . . . , kJ) and l = (ℓ1, . . . , ℓJ) are vectors in (Z≥0)J.
+• u = (u1, . . . , uJ), v = (v1, . . . , vJ), x = (x1, . . . , xJ) and y = (y1, . . . , yJ) are
+vectors in RJ.
+• z = (z1, . . . , zJ) = x + iy and ¯z = (z1, . . . , zJ) = x − iy are vectors in CJ.
+• k! := k1! · · · kJ! and K(k) := k1 + · · · + kJ.
+• xk := xk1
+1 · · ·xkJ
+J .
+• x · u = �J
+j=1 xjuj, ||z|| =
+��J
+j=1 |zj|2 =
+��J
+j=1(x2
+j + y2
+j).
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+5
+2. Estimates on random model
+We define the random vector
+L(σ, X) =
+�
+log |L1(σ, X)|, . . . , log |LJ(σ, X)|, arg L1(σ, X), . . . , arg LJ(σ, X)
+�
+for σ > 1
+2, where each Lj(σ, X) is defined by the product
+(2.1)
+Lj(σ, X) =
+�
+p
+d
+�
+i=1
+�
+1 − αj,i(p)X(p)
+pσ
+�−1
+and {X(p)}p is a sequence of independent random variables, indexed by the prime
+numbers, and uniformly distributed on the unit circle {z ∈ C : |z| = 1}. The product
+converges almost surely for σ > 1
+2 by Kolmogorov’s three series theorem.
+Define a probability measure
+(2.2)
+Φrand
+T
+(B) := P(L(σT, X) ∈ B)
+for a Borel set B in R2J. By [4, Theorem 2.3] we have
+ΦT (RT) = Φrand
+T
+(RT ) + O((log T)(θ−1)/2 log log T)
+for 0 < θ < 1
+2. It means that the distribution of L(σT + it) is well approximated by the
+distribution of its random model L(σT , X) when 0 < θ < 1
+2. Thus, Theorem 1.2 is an
+immediate consequence of the following theorem.
+Theorem 2.1. Let 0 < θ < 1
+2. Assume assumptions A1–A5 for L1, . . . , LJ. Then there
+exist constants ǫ, κ > 0 and a sequence {bk,l} of real numbers such that
+Φrand
+T
+(RT) =
+�
+K(k+l)≤ǫ log log T
+bk,l
+J�
+j=1
+1
+�
+ψj,T
+kj+ℓj
+×
+J
+�
+j=1
+� � bj
+aj
+e−πu2Hkj(√πu)du
+� dj
+cj
+e−πv2Hℓj(√πv)dv
+�
++ O
+�
+1
+(log T)κ
+�
+.
+Moreover, b0,0 = 1, bk,l = 0 if K(k + l) = 1 and bk+l = O(δ−K(k+l)
+0
+) for some δ0 > 0
+and all k, l.
+In [4, Section 7] we find that the measure Φrand
+T
+is absolutely continuous and it has
+a density function HT(u, v) such that
+(2.3)
+Φrand
+T
+(RT) =
+��
+RT
+HT(u, v)dudv.
+Hence, Theorem 2.1 follows from (2.3) and the following proposition, which upgrades
+[4, Lemma 7.4].
+
+6
+YOONBOK LEE
+Proposition 2.2. Let 0 < θ < 1
+2. Assume assumptions A1–A5 for L1, . . . , LJ. There
+exist constants ǫ, κ > 0 and a sequence {bk,l} of real numbers such that
+HT(u, v) =
+�
+K(k+l)≤ǫ log log T
+bk,l
+J�
+j=1
+1
+π
+�
+ψj,T
+kj+ℓj+2e
+−
+u2
+j +v2
+j
+ψj,T Hkj
+�
+uj
+�
+ψj,T
+�
+Hℓj
+�
+vj
+�
+ψj,T
+�
++ O
+�
+1
+(log T)κ
+�
+.
+Moreover, b0,0 = 1, bk,l = 0 if K(k + l) = 1 and bk+l = O(δ−K(k+l)
+0
+) for some δ0 > 0
+and all k, l.
+To prove Proposition 2.2, it requires to understand the Fourier transform
+�Φrand
+T
+(x, y) :=
+�
+R2J e2πi(x·u+y·v)dΦrand
+T
+(u, v)
+for x, y ∈ RJ. By the definition of Φrand
+T
+in (2.2), we have
+�Φrand
+T
+(x, y) = E
+�
+exp
+�
+2πi
+J
+�
+j=1
+�
+xj log |Lj(σT , X)| + yj arg Lj(σT, X)
+�
+��
+.
+By assumptions A1 and A5 we see that
+(2.4)
+βLj(pk) = 1
+k
+d
+�
+i=1
+αj,i(p)k.
+By (2.4) and (2.1) we have
+log Lj(σ, X) =
+�
+p
+∞
+�
+k=1
+βLj(pk)X(p)k
+pk��
+.
+Define
+(2.5)
+gj,p(σ) :=
+∞
+�
+k=1
+βLj(pk)X(p)k
+pkσ
+,
+then we have
+(2.6)
+�Φrand
+T
+(x, y) =
+�
+p
+ϕp,σT (x, y),
+where
+ϕp,σ(x, y) := E
+�
+exp
+�
+2πi
+J
+�
+j=1
+�
+xjRe (gj,p(σ)) + yjIm (gj,p(σ))
+�
+��
+for each prime p. Let z = (z1, . . . , zJ) = x + iy, then we find that
+ϕp,σ(x, y) = E
+� J�
+j=1
+eπizjgj,p(σ)eπizjgj,p(σ)
+�
+.
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+7
+By expanding the 2J exponential functions into power series we obtain
+ϕp,σ(x, y) =
+�
+k,l∈(Z≥0)J
+(πi)K(k+l)zkzl
+k!l!
+E
+�
+J�
+j=1
+gj,p(σ)kjgj,p(σ)
+ℓj
+�
+with notations for vectors in the end of Section 1. It is easy to see that the expectation
+(2.7)
+Ap,σ(k, l) := E
+�
+J�
+j=1
+gj,p(σ)kjgj,p(σ)
+ℓj
+�
+satisfies Ap,σ(0, 0) = 1 and Ap,σ(0, k) = Ap,σ(k, 0) = 0 for k ̸= 0. Thus, we obtain
+(2.8)
+ϕp,σ(x, y) = 1 + Rp,σ(z),
+where
+(2.9)
+Rp,σ(z) :=
+�
+k̸=0
+�
+l̸=0
+(πi)K(k+l)zkzl
+k!l!
+Ap,σ(k, l).
+Hence, by (2.6) and (2.8) we have
+(2.10)
+�Φrand
+T
+(x, y) =
+�
+p
+(1 + Rp,σT (z)).
+To compute the product in (2.10), it requires the following lemma.
+Lemma 2.3. There exists a constant δ1 > 0 such that
+|Rp,σT (z)| ≤ 1
+2
+for every prime p and ||z|| ≤ δ1.
+See Section 3.1 for a proof. By Lemma 2.3 we have
+�Φrand
+T
+(x, y) = exp
+� �
+p
+log(1 + Rp,σT (z))
+�
+= exp
+� �
+p
+∞
+�
+m=1
+(−1)m−1
+m
+Rp,σT (z)m
+�
+(2.11)
+for ||z|| ≤ δ1. By (2.9) the sum �
+p
+�∞
+m=1
+(−1)m−1
+m
+Rp,σ(z)m has a power series represen-
+tation in z1, . . . , zJ, z1, . . . , zJ, so let Bσ(k, l) be the coefficients such that
+(2.12)
+�
+k̸=0
+�
+l̸=0
+Bσ(k, l)zkzl =
+�
+p
+∞
+�
+m=1
+(−1)m−1
+m
+Rp,σ(z)m.
+Define In,σ(z) for each n ≥ 2 by the sum of the degree n terms in the above sum, i.e.,
+(2.13)
+In,σ(z) :=
+�
+k,l̸=0
+K(k+l)=n
+Bσ(k, l)zkzl.
+
+8
+YOONBOK LEE
+We see that In,σ(z) is a homogeneous polynomial in x1, . . . , xJ, y1, . . . , yJ of degree n,
+and that
+(2.14)
+�Φrand
+T
+(x, y) = exp
+� ∞
+�
+n=2
+In,σT (z)
+�
+for ||z|| ≤ δ1 by (2.11)–(2.13). We find an asymptotic formula for In,σT (z) as T → ∞
+in the following lemma.
+Lemma 2.4. There are complex numbers Cj1,j2 such that
+(2.15)
+I2,σT (z) = −π2
+J
+�
+j=1
+ψj,T|zj|2 +
+J
+�
+j1,j2=1
+Cj1,j2zj1zj2 + O
+�log log T
+(log T)θ
+�
+for ||z|| ≤ δ1, where ψj,T is defined in (1.4) and Cj1,j2 = Cj2,j1. For n ≥ 3, there is a
+constant C = CJ,d,η > 0 such that
+|In,σ(z)| ≤ Cn||z||n
+for σ ≥ 1
+2 and
+|In,σT (z) − In,1/2(z)| ≤ Cn||z||n
+(log T)θ .
+See Section 3.2 for a proof. Define
+(2.16)
+QT(z) := −π2
+J
+�
+j=1
+ψj,T|zj|2,
+(2.17)
+I2(z) :=
+J
+�
+j1,j2=1
+Cj1,j2zj1zj2
+and
+(2.18)
+In(z) := In,1/2(z)
+for n > 2. By (2.17) and the Cauchy-Schwarz inequality we obtain
+|I2(z)| ≤ J(max
+j1,j2 |Cj1,j2|)||z||2.
+By this inequality, (2.18) and Lemma 2.4 we have
+(2.19)
+|In(z)| ≤ 2−n
+for n ≥ 2 and ||z|| ≤ δ2, where
+(2.20)
+δ2 := min
+�
+δ1, 1
+2C ,
+1
+2
+�
+J maxj1,j2 |Cj1,j2|
+�
+.
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+9
+It follows from (2.14), Lemma 2.4 and (2.16)–(2.19) that
+�Φrand
+T
+(x, y) = exp
+�
+QT(z) +
+∞
+�
+n=2
+In(z) + O
+�log log T
+(log T)θ
+��
+= eQT (z)
+� ∞
+�
+r=0
+1
+r!
+�
+∞
+�
+n=2
+In(z)
+�r
++ O
+�log log T
+(log T)θ
+��
+(2.21)
+for ||z|| ≤ δ2. Note that each In(z) is a homogeneous polynomial in x1, . . . , xJ, y1, . . . , yJ
+of degree n and does not depend on T. Since the sum �∞
+r=0
+1
+r!
+� �∞
+n=2 In(z)
+�r is a power
+series in x and y, we let {bk,l} be a sequence of complex numbers such that
+(2.22)
+G(x, y) :=
+�
+k,l
+(2πi)K(k+l)bk,lxkyl =
+∞
+�
+r=0
+1
+r!
+�
+∞
+�
+n=2
+In(z)
+�r
+.
+Then the bk,l satisfy the following properties.
+Lemma 2.5. Let δ3 be a constant satisfying 0 < δ3 <
+π
+√
+J δ2, then bk,l is a real number
+and
+(2.23)
+|bk,l| ≤
+√e
+δK(k+l)
+3
+for every k, l. In particular, b0,0 = 1 and bk,l = 0 if K(k + l) = 1.
+See Section 3.3 for a proof. The infinite sum over k, l in (2.22) can be approximated
+by its partial sum. We shall prove a quantitative version. Let ǫ > 0. By (2.22) and
+(2.19) we have
+����
+�
+K(k+l)>ǫ log log T
+(2πi)K(k+l)bk,lxkyl
+���� ≤
+∞
+�
+r=1
+1
+r!
+�
+n1,...,nr≥2
+n1+···+nr>ǫ log log T
+�1
+2
+�n1+···+nr
+≤
+∞
+�
+r=1
+1
+r!
+�
+m>ǫ log log T
+1
+2m
+�
+n1,...,nr≥2
+n1+···+nr=m
+1
+for ||z|| ≤ δ2. We substitute nj by n′
+j + 2 for j = 1, . . . , r in the last sum, then the last
+sum equals to the number of nonnegative integers n′
+1, . . . , n′
+r such that n′
+1 + . . . + n′
+r =
+m − 2r, which equals to
+�m−r−1
+r−1
+�
+. Thus, the above sum is
+≤
+∞
+�
+r=1
+1
+r!
+�
+m>ǫ log log T
+1
+2m
+�m − r − 1
+r − 1
+�
+≤
+∞
+�
+r=1
+1
+r!
+�
+m>ǫ log log T
+1
+2m
+mr−1
+(r − 1)!
+≤
+�
+m>ǫ log log T
+1
+2m
+∞
+�
+n=0
+mn
+(n!)2 ≤
+�
+m>ǫ log log T
+1
+2m
+�
+∞
+�
+n=0
+√mn
+n!
+�2
+=
+�
+m>ǫ log log T
+e2√m
+2m
+≤
+�
+m>ǫ log log T
+�2
+3
+�m
+≤ 3
+�2
+3
+�ǫ log log T
+≪
+1
+(log T)κ
+
+10
+YOONBOK LEE
+with a constant κ ≤ ǫ log 3
+2. It follows from these estimates, (2.21), (2.22) and Lemma
+2.5 we obtain the following proposition.
+Proposition 2.6. Let δ2 be the constant defined in (2.20). Let κ and ǫ be constants
+such that 0 < κ < θ and κ ≤ ǫ log 3
+2. Let {bk,l} be a sequence of real numbers defined
+by its generating series (2.22). Then
+�Φrand
+T
+(x, y) = eQT (z)
+�
+�
+K(k+l)≤ǫ log log T
+(2πi)K(k+l)bk,lxkyl + O
+�
+1
+(log T)κ
+��
+holds for ||z|| ≤ δ2.
+We are ready to prove Proposition 2.2. The density function HT(u, v) of the measure
+Φrand
+T
+is the inverse Fourier transform of �Φrand
+T
+, so that
+HT(u, v) =
+�
+RJ
+�
+RJ
+�Φrand
+T
+(x, y)e−2πi(x·u+y·v)dxdy.
+Let δ4 be a constant such that 0 < δ4 ≤ min{δ2, δ3
+4π}. By Lemma 7.1 and (7.14) in [4]
+we find that
+HT(u, v) =
+��
+||z||≤δ4
+�Φrand
+T
+(x, y)e−2πi(x·u+y·v)dxdy + O
+�
+1
+(log T)κ
+�
+for some κ > 0. See the proof of [4, Lemma 7.4] for a detail.
+By Proposition 2.6 we have
+HT(u, v) =
+�
+K(k+l)≤ǫ log log T
+(2πi)K(k+l)bk,l
+��
+||z||≤δ4
+eQT (z)−2πi(x·u+y·v)xkyldxdy+O
+�
+1
+(log T)κ
+�
+for some ǫ, κ > 0. Let ξmin = minj≤J ξj > 0, then we have
+����
+��
+||z||≥δ4
+eQT (z)−2πi(x·u+y·v)xkyldxdy
+���� ≤
+��
+||z||≥δ4
+e−π2ξminθ log log T||z||2||z||K(k+l)dxdy
+≪
+� ∞
+δ4
+e−(π2ξminθ log log T)r2rK(k+l)+2J−1dr
+≪
+1
+(π2ξminθ log log T)
+K(k+l)
+2
++J
+� ∞
+πδ4
+√ξminθ log log T
+e−r2rK(k+l)+2J−1dr
+by the change of variables to the polar coordinates. By the Cauchy-Schwarz inequality
+we have
+� ∞
+X
+e−r2rMdr ≤
+�� ∞
+X
+e−r2rdr
+� ∞
+0
+e−r2r2M−1dr =
+�
+(M − 1)!
+2
+e− 1
+2 X2.
+Hence, it follows from Lemma 2.5 and the above estimations that
+HT(u, v) =
+�
+K(k+l)≤ǫ log log T
+(2πi)K(k+l)bk,l
+�
+RJ
+�
+RJ eQT (z)−2πi(x·u+y·v)xkyldxdy
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+11
++ O
+�
+1
+(log T)
+1
+2π2δ2
+4ξminθ
+�
+K(k+l)≤ǫ log log T
+�2π
+δ3
+�K(k+l) �
+(K(k + l) + 2J − 2)!
+(π2ξminθ log log T)
+K(k+l)
+2
++J
+�
++ O
+�
+1
+(log T)κ
+�
+.
+By Stirling’s formula the k, l-sum in the above O-term is
+≪
+�
+K(k+l)≤ǫ log log T
+�2π
+δ3
+�K(k+l)
+1
+(π2ξminθ log log T)
+K(k+l)
+2
++J
+�2ǫ log log T
+e
+� K(k+l)
+2
++J− 3
+4
+≪
+�
+k,l
+�
+2
+√
+2ǫ
+δ3
+√ξminθe
+�K(k+l)
+≤
+�
+k,l
+�1
+2
+�K(k+l)
+= 22J,
+provided that 0 < ǫ ≤
+1
+32δ2
+3ξminθe. With this choice of ǫ, we have
+HT(u, v) =
+�
+K(k+l)≤ǫ log log T
+(2πi)K(k+l)bk,l
+�
+RJ
+�
+RJ eQT (z)−2πi(x·u+y·v)xkyldxdy
++ O
+�
+1
+(log T)κ
+�
+for some κ > 0
+It remains to calculate the above integral. We first write it as repeated integrals
+�
+RJ
+�
+RJ eQT (z)−2πi(x·u+y·v)xkyldxdy
+=
+J
+�
+j=1
+�
+R
+�
+R
+e−ψj,T π2(x2
+j+y2
+j )−2πi(xjuj+yjvj)x
+kj
+j y
+ℓj
+j dxjdyj
+=
+J
+�
+j=1
+�
+R
+e−ψj,T π2x2
+j−2πixjujx
+kj
+j dxj
+�
+R
+e−ψj,T π2y2
+j −2πiyjvjy
+ℓj
+j dyj.
+Each integral can be written in terms of the Hermite polynomials defined in (1.2). Since
+�
+R
+e−ψπ2x2−2πixuxkdx =
+1
+(−2πi)k
+dk
+duk
+�
+R
+e−ψπ2x2−2πixudx
+=
+1
+(−2πi)k
+dk
+duk
+1
+√πψe− u2
+ψ
+=
+1
+(2πi)k√π√ψ
+k+1e− u2
+ψ Hk
+� u
+√ψ
+�
+,
+we have
+�
+RJ
+�
+RJ eQT (z)−2πi(x·u+y·v)xkyldxdy
+
+12
+YOONBOK LEE
+=
+J�
+j=1
+1
+π(2πi)kj+ℓj�
+ψj,T
+kj+ℓj+2e
+−
+u2
+j +v2
+j
+ψj,T Hkj
+�
+uj
+�
+ψj,T
+�
+Hℓj
+�
+vj
+�
+ψj,T
+�
+.
+Thus, we have
+HT(u, v) =
+�
+K(k+l)≤ǫ log log T
+bk,l
+J�
+j=1
+1
+π
+�
+ψj,T
+kj+ℓj+2e
+−
+u2
+j +v2
+j
+ψj,T Hkj
+�
+uj
+�
+ψj,T
+�
+Hℓj
+�
+vj
+�
+ψj,T
+�
++ O
+�
+1
+(log T)κ
+�
+for some ǫ, κ > 0. This completes the proof of Proposition 2.2.
+3. Proofs of lemmas
+We prove Lemma 2.3 in Section 3.1, Lemma 2.4 in Section 3.2 and Lemma 2.5 in
+Section 3.3. In the proofs, we need the inequalities
+(3.1)
+|βLj(pk)| ≤ d
+kpkη
+for k ≥ 1,
+(3.2)
+|βLj(pk)| ≤ 1
+k
+d
+�
+i=1
+|αj,i(p)|k ≤ p(k−2)η
+k
+d
+�
+i=1
+|αj,i(p)|2
+for k ≥ 2
+and
+(3.3)
+|βLj(p)|2 ≤
+�
+d
+�
+i=1
+|αj,i(p)|
+�2
+≤ d
+d
+�
+i=1
+|αj,i(p)|2,
+which follows by (2.4) and assumpion A1.
+3.1. Proof of Lemma 2.3. By (2.5) and (3.1) there is a constant C1 := C1,d,η > 0
+such that
+(3.4)
+|gj,p(σT)| ≤
+∞
+�
+k=1
+d
+k
+pkη
+p
+k
+2 ≤
+C1
+p
+1
+2 −η
+for every prime p and j = 1, . . . , J. By (2.7), (2.9) and (3.4) we obtain
+|Rp,σT (z)| ≤
+�
+k̸=0
+�
+l̸=0
+1
+k!l!
+�
+π||z|| C1
+p
+1
+2−η
+�K(k+l)
+=
+�
+exp
+�
+J πC1||z||
+p
+1
+2−η
+�
+− 1
+�2
+.
+Thus, there exists a constant C2 := C2,d,J,η > 0 such that
+|Rp,σT (z)| ≤
+C2
+p1−2η ||z||2 ≤
+C2
+21−2�� ||z||2
+for ||z|| ≤ 1 and every prime p. Therefore, there exists a constant δ1 > 0 such that
+|Rp,σT (z)| ≤ 1
+2
+for ||z|| ≤ δ1 and every prime p.
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+13
+3.2. Proof of Lemma 2.4. We first find an useful expression
+(3.5)
+In,σ(z) = (πi)n
+�
+1≤m≤n/2
+(−1)m−1
+m
+�
+k1,...,km,l1,...,lm̸=0
+K(k1+···+km+l1+···+lm)=n
+zk1+···+kmzl1+···+lm
+k1! · · ·km!l1! · · ·lm!
+×
+�
+p
+Ap,σ(k1, l1) · · ·Ap,σ(km, lm)
+by (2.9), (2.12) and (2.13). Here, the sum over m is 1 ≤ m ≤ n/2 because
+n = K(k1 + · · · + km + l1 + · · · + lm) ≥ 2m
+for k1, . . . , km, l1, . . . , lm ̸= 0.
+The asymptotic (2.15) of I2,σT (z) is known before. See (7.16) of [4, Lemma 7.3]. We
+next prove
+(3.6)
+Cj1,j2 = Cj2,j1.
+We have
+(3.7)
+Ap,σ(k, l) = Ap,σ(l, k)
+by (2.7). By (3.5) we also have
+(3.8)
+I2,σ(z) = I2,σ(z).
+So we obtain (3.6) by (2.15) and (3.8).
+For the case n > 2, we observe that Ap,σ(k, l) for a real σ can be extended to an
+analytic function in a complex variable s via
+(3.9)
+Ap,s(k, l) = E
+�
+J�
+j=1
+� ∞
+�
+k=1
+βLj(pk)X(p)k
+pks
+�kj� ∞
+�
+k=1
+βLj(pk)X(p)k
+pks
+�ℓj�
+.
+This observation essentially leads us to prove the following lemma.
+Lemma 3.1. Let η be the constant in assumption A1 and assume K(k1 + · · · + km +
+l1 + · · · + lm) = n ≥ 3. The Dirichlet series
+f(s) :=
+�
+p
+Ap,s(k1, l1) · · ·Ap,s(km, lm)
+is absolutely convergent for Re(s) ≥
+5+2η
+12 . Moreover, there exists a constant C3 =
+C3,J,d,η > 0 such that
+|f(s)| ≤ Cn
+3
+for Re(s) ≥ 5+2η
+12
+and
+|f(σT) − f( 1
+2)| ≤
+Cn
+3
+(log T)θ .
+
+14
+YOONBOK LEE
+Proof. We first show that there is a constant C4 > 0 such that
+|f(s)| ≤ Cn
+4
+for Re(s) ≥ 5+2η
+12 . By (3.9) we find that
+|Ap,s(k, l)| ≤
+� ∞
+�
+k=1
+maxj≤J |βLj(pk)|
+pkRe(s)
+�K(k+l)
+.
+Thus, we have
+|f(s)| ≤
+�
+p
+�
+∞
+�
+k=1
+maxj≤J |βLj(pk)|
+pkRe(s)
+�n
+≤ 2n �
+p
+�maxj≤J |βLj(p)|
+pRe(s)
+�n
++ 2n �
+p
+� ∞
+�
+k=2
+maxj≤J |βLj(pk)|
+pkRe(s)
+�n
+.
+(3.10)
+The first sum on the right hand side of (3.10) is
+�
+p
+�
+maxj≤J |βLj(p)|
+�n
+pnRe(s)
+≤
+�
+p
+(dpη)n−2�
+maxj≤J d �d
+i=1 |αj,i(p)|2�
+pnRe(s)
+≤ dn−1 �
+p
+�J
+j=1
+�d
+i=1 |αj,i(p)|2
+p1+ε
+≤ Cn
+5
+for Re(s) ≥ 5+2η
+12
+by (3.1) and (3.3), where ε = 1
+4 − η
+2 > 0 and
+C5 := max
+�
+d,
+�
+p
+�J
+j=1
+�d
+i=1 |αj,i(p)|2
+p1+ε
+�
+.
+Note that the last p-sum is convergent by assumption A3 and a partial summation.
+The second sum on the right hand side of (3.10) is
+�
+p
+� ∞
+�
+k=2
+maxj≤J |βLj(pk)|
+pkRe(s)
+�n
+≤
+�
+p
+�
+∞
+�
+k=2
+maxj≤J
+�d
+i=1 |αj,i(p)|2
+kpkRe(s)−(k−2)η
+�n
+≤
+�
+p
+�maxj≤J
+�d
+i=1 |αj,i(p)|2
+p2Re(s)
+1
+2
+1
+1 −
+1
+pRe(s)−η
+�n
+≤
+�1
+2
+1
+1 −
+1
+p
+5
+12 (1−2η)
+�n �
+p
+(dp2η)n−1 maxj≤J
+�d
+i=1 |αj,i(p)|2
+p2nRe(s)
+≤
+�1
+2
+1
+1 −
+1
+p
+5
+12 (1−2η)
+�n
+dn−1 �
+p
+�J
+j=1
+�d
+i=1 |αj,i(p)|2
+p1+6ε
+≤ Cn
+6
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+15
+for Re(s) ≥ 5+2η
+12
+by (3.2), where
+C6 := 1
+2
+1
+1 −
+1
+p
+5
+12 (1−2η)
+max
+�
+d,
+�
+p
+�J
+j=1
+�d
+i=1 |αj,i(p)|2
+p1+6ε
+�
+.
+We choose C4 = 2(C5 + C6), then we have
+(3.11)
+|f(s)| ≤ Cn
+4
+for Re(s) ≥ 5+2η
+12 . One can easily see in the above estimations that f(s) is absolutely
+convergent for Re(s) ≥ 5+2η
+12 .
+Let ε1 = 1
+2 − 5+2η
+12
+> 0. Since
+f(σT) − f( 1
+2) =
+� σT
+1/2
+f ′(u)du =
+� σT
+1/2
+1
+2πi
+�
+|z−u|=ε1
+f(z)
+(z − u)2dzdu,
+we obtain
+(3.12)
+|f(σT) − f( 1
+2)| ≤ (σT − 1
+2) 1
+ε1
+sup
+Re(z)≥ 1
+2 −ε1
+|f(z)| ≤
+Cn
+4
+ε1(log T)θ
+by (3.11). Let C3 = C4/ε1 > C4, then (3.11) and (3.12) imply both inequalities in the
+lemma.
+□
+Therefore by Lemma 3.1, (3.5) and Stirling’s formula we have
+|In,σ(z)| ≤ ||z||n(πC3)n �
+m≤n/2
+1
+m
+�
+K(k1+···+km+l1+···+lm)=n
+1
+k1! · · · km!l1! · · ·lm!
+= ||z||n(πC3)n �
+m≤n/2
+1
+m
+(2mJ)n
+n!
+≤ ||z||n(JπC3)nnn
+n!
+≤ ||z||n(JπC3e)n
+for σ ≥ 5+2η
+12
+and n > 2. Similarly, we have
+|In,σT (z) − In,1/2(z)| ≤ ||z||n(JπC3e)n
+(log T)θ
+for n > 2. Therefore, Lemma 2.4 holds with a constant
+(3.13)
+C = JπC3e.
+3.3. Proof of Lemma 2.5. We first consider G(x, y) in (2.22) as a function in complex
+variables x1, . . . , xJ, y1, . . . , yJ. We replace xj by
+xj
+2πi and yj by
+yj
+2πi for j = 1, . . . , J in
+(2.22), then we obtain that
+(3.14)
+�
+k,l
+bk,lxkyl =
+∞
+�
+r=0
+1
+r!
+� ∞
+�
+n=2
+In(z)(2πi)−n
+�r
+.
+
+16
+YOONBOK LEE
+Now we consider x1, . . . , xJ, y1, . . . , yJ as real variables. By (3.5) and (3.7) we have
+In,σ(z)(2πi)−n = In,σ(z)(2πi)−n,
+which implies that In,σ(z)(2πi)−n is a polynomial in real variables x1, . . . , xJ, y1, . . . , yJ
+with real coefficients. Since In(z)(2πi)−n is also a homogeneous polynomial in x1, . . . , xJ,
+y1, . . . , yJ of degree n with real coefficients, we obtain by comparing coefficients in (3.14)
+that bk,l ∈ R, b0,0 = 1 and bk,l = 0 for K(k + l) = 1.
+It remains to prove the inequality (2.23). Again we consider G(x, y) defined in (2.22)
+as an analytic function in complex variables x1, . . . , xJ, y1, . . . , yJ. Assume that
+sup{|x1|, . . . , |xJ|, |y1|, . . . , |yJ|} ≤
+δ2
+2
+√
+J
+.
+Then we see that
+|I2(z)| ≤
+J
+�
+j1,j2=1
+|Cj1,j2| δ2
+2
+4J ≤ 1
+16
+by (2.17) and (2.20). For n ≥ 3 we have
+|In(z)| ≤
+�δ2πC3
+√
+J
+�n �
+m≤n/2
+1
+m
+�
+K(k1+···+km+l1+···+lm)=n
+1
+k1! · · ·km!l1! · · ·lm!
+≤ (δ2
+√
+JπC3e)n ≤ (δ2C)n ≤ 2−n
+by (2.18), (2.20), (3.5), (3.13) and Lemma 3.1. Thus,
+|G(x, y)| ≤
+∞
+�
+r=0
+1
+r!
+�
+∞
+�
+n=2
+|In(z)|
+�r
+≤
+∞
+�
+r=0
+1
+r!2−r = √e.
+Let 0 < δ3
+2π = δ′
+3 <
+δ2
+2
+√
+J . Since
+bk,l =
+1
+(2πi)K(k+l)+2J
+�
+|x1|=δ′
+3
+· · ·
+�
+|xJ|=δ′
+3
+�
+|y1|=δ′
+3
+· · ·
+�
+|yJ|=δ′
+3
+G(x, y)
+xkyl
+dyJ
+yJ
+· · · dy1
+y1
+dxJ
+xJ
+· · · dx1
+x1
+by Cauchy’s integral formula, we obtain
+|bk,l| ≤
+√e
+(2πδ′
+3)K(k+l) =
+√e
+δK(k+l)
+3
+.
+Acknowledgements
+This work has been supported by the National Research Foundation of Korea (NRF)
+grant funded by the Korea government (MSIP) (No. 2019R1F1A1050795).
+
+CENTRAL LIMIT THEOREM OF L-FUNCTIONS
+17
+References
+[1] H. Cram´er, Random variables and probability distributions, 3rd edition, Cambridge University
+Press, 1970.
+[2] J. Ha and Y. Lee, The a-values of the Riemann zeta function near the critical line, J. Math. Anal.
+Appl. 464, (2018), 838–863.
+[3] D. Hejhal, On Euler products and multi-variate Gaussians, C. R. Acad. Sci. Paris, Ser. I 337
+(2003), 223–226.
+[4] Y. Lamzouri and Y. Lee, The number of zeros of linear combinations of L-functions near the
+critical line, to appear J. Anal. Math. Preprint available at arXiv:2010.10490.
+[5] Y. Lee, An asymptotic expansion of Selberg’s central limit theorem near the critical line, J. Number
+Theory 236 (2022), 323–333.
+[6] M. Radziwi�l�l and K. Soundararajan, Selberg’s central limit theorem for log |ζ( 1
+2 + it)|, Enseign.
+Math. 63 (2017), 1–19.
+[7] A. Selberg, Old and new conjectures and results about a class of Dirichlet series, Bombieri, E.
+(ed.) et al., Proceedings of the Amalfi conference on analytic number theory, held at Maiori,
+Amalfi, Italy, from 25 to 29 September, 1989. Salerno: Universit´a di Salerno, 367–385 (1992) =
+Collected Papers, vol. II, 47–63, Springer, 1991.
+[8] K.M. Tsang, The distribution of the values of the Riemann zeta-function, ProQuest LLC, Ann
+Arbor, MI, 1984, Thesis (Ph.D.)-Princeton University.
+Department of Mathematics, Research Institute of Basic Sciences, Incheon Na-
+tional University, 119 Academy-ro, Yeonsu-gu, Incheon, 22012, Korea
+Email address: leeyb@inu.ac.kr, leeyb131@gmail.com
+

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+arXiv:2301.01473v1  [math.CO]  4 Jan 2023
+State Transfer in Complex Quantum Walks
+Antonio Acuaviva1, Ada Chan2, Summer Eldridge3, Chris Godsil4, Matthew How-Chun-Lun5,
+Christino Tamon6, Emily Wright7, and Xiaohong Zhang8
+1Department of Mathematics, Universidad Complutense de Madrid
+2Department of Mathematics and Statistics, York University
+3Department of Mathematics, University of Toronto
+4Department of Combinatorics and Optimization, University of Waterloo
+5Department of Mathematics, McMaster University
+6Department of Computer Science, Clarkson University
+7Department of Mathematics, Queen’s University
+8Centre de recherches mathématiques, Université de Montréal
+January 5, 2023
+Abstract
+Given a graph with Hermitian adjacency matrix 퐻, perfectstate transfer occurs fromvertex 푎 to vertex
+푏 if the (푏, 푎)-entryof the unitary matrix exp(−푖퐻푡) has unit magnitudefor some time 푡. This phenomenon
+is relevant for information transmission in quantum spin networks and is known to be monogamous under
+real symmetric matrices. We prove the following results:
+• For oriented graphs (whose nonzero weights are ±푖), the oriented 3-cycle and the oriented edge
+are the only graphs where perfect state transfer occurs between every pair of vertices. This settles
+a conjecture of Cameron et al. [1]. On the other hand, we construct an infinite family of oriented
+graphs with perfect state transfer between any pair of vertices on a subset of size four.
+• There are infinite families of Hermitian graphs with one-way perfect state transfer, where perfect
+state transfer occurs without periodicity. In contrast, perfect state transfer implies periodicity when-
+ever the adjacency matrix has algebraic entries (see Godsil [2]).
+• There are infinite families with non-monogamous pretty good state transfer in rooted graph prod-
+ucts. In particular, we generalize known results on double stars (due to Fan and Godsil [3]) and
+on paths with loops (due to Kempton, Lippner and Yau [4]). The latter extends the experimental
+observation of quantum transport (made by Zimborás et al. [5]) and shows non-monogamouspretty
+good state transfer can occur amongst distant vertices.
+1
+Introduction
+Given a graph 푋 = (푉 , 퐸) with adjacency matrix 퐴, a continuous-time quantum walk on 푋 is defined by the
+time-dependent unitary matrix 푈(푡) = 푒−푖퐴푡. This natural quantum generalization of continuous-time ran-
+dom walks is important for designing quantum algorithms. Childs et al. [6] showed that a continuous-time
+quantum walk algorithm provides an exponential time speedup for an explicit search problem on graphs.
+1
+
+Subsequently, Childs [7] showed that continuous-time quantum walk is a universal model of quantum com-
+putation.
+Our focus in this paper is motivated by Bose [8] who studied quantum communication via continuous-
+time quantum walk on graphs. We say that there is pretty good state transfer in a graph 푋 from vertex 푎 to
+vertex 푏 if for any 휖 > 0, there is a time 푡 so that ‖‖푈(푡)푒푎 − 훾푒푏‖‖ ≤ 휖 where 훾 is a phase factor. Here, 푒푎
+denotes the unit vector with 1 at position 푎 and 0 elsewhere; similarly for 푒푏. If 휖 = 0 is achievable, we say
+there is perfect state transfer in 푋 from 푎 to 푏 at time 푡.
+Kay [9] proved a monogamy property for perfect state transfer on graphs with real symmetric adjacency
+matrices: if there is perfect state transfer from 푎 to 푏 and from 푎 to 푐 then 푏 = 푐. In contrast, Cameron
+et al. [1] showed that there are oriented graphs (whose adjacency matrices are Hermitian with ±푖 nonzero
+entries) where state transfer occurs between every pair of vertices. This latter property is called universal
+state transfer. Their primary examples are oriented cycles of prime order with universal pretty good state
+transfer. A notable exception is the oriented 3-cycle which exhibits universal perfect state transfer.
+It was conjectured in [1] that the oriented 퐾2 and 3-cycle are the only oriented graph with universal perfect
+state transfer. We prove their conjecture in this work. This confirms that universal perfect state transfer is
+an extremely rare phenomenon in oriented graphs. On the other hand, there are known infinite families of
+graphs with universal perfect state transfer but with adjacency matrices that are Hermitian matrices with no
+restriction on the entries (see Connelly et al. [10]). We call these Hermitian graphs.
+Godsil and Lato [11] proved a strong characterization of perfect state tranfer in oriented graphs and
+observed that perfect state transfer always implies periodicity (by the Gelfond-Schneider theorem). In fact,
+Godsil [2] had observed the latter property holds for any adjacency matrix with algebraic entries. Our next
+observation shows that the latter assumption is necessary to guarantee periodicity. We construct the first
+infinite family of Hermitian graphs with one-way perfect state transfer, where perfect state transfer occurs
+without periodicity. These examples also exhibit a one-time perfect state transfer property where perfect state
+transfer occurs at a single unique time (and never to repeat again).
+Godsil and Lato [11] also introduced a relaxation of universal perfect state transfer called multiple perfect
+state transfer. We say a graph 푋 has multiple state transfer on a subset 푆 ⊂ 푉 (푋) of vertices, with |푆| ≥ 3,
+if state transfer occurs between every pair of vertices of 푆. An explicit example of a 8-vertex circulant with
+multiple perfect state tranfer was given in [11], but it was not clear if there are more examples sharing the
+same properties. We construct the first infinite family of oriented graphs with multiple perfect state transfer
+(which contains the aforementioned 8-vertex circulant as a special case). This shows that, unlike universal
+perfect state transfer, multiple perfect state transfer is not an extremely rare phenomenon.
+It is known that perfect state transfer is closed under the Cartesian graph product. In this work, under mild
+assumptions, we show that multiple state transfer is closed under the rooted graph product (see Godsil and
+McKay [12]). First, we prove a complete characterization of pretty good state transfer on the rooted product
+of the oriented 3-cycle with stars 퐾1,푚. This generalizes a result of Fan and Godsil [3] on the double stars.
+Next, we consider rooted product with single-looped paths instead of stars. Let 푋 be a 푛-vertex circulant
+with universal perfect state transfer and let 푃 훾
+푚 be a 푚-vertex path with a self-loop of weight 훾 at one of its
+endpoints. We prove that the rooted product 푋◦푃 훾
+푚 has multiple pretty good state transfer between every
+pair of vertices with self-loop provided 훾 is transcendental. This generalizes a result of Kempton, Lippner
+and Yau [4] and shows the power of loops to facilitate multiple state transfer among distant vertices. In the
+special case when 푋 is the oriented 3-cycle, our result strengthens the experimental observations in Zimborás
+et al. [5] (with the help of self-loops).
+2
+
+2
+Preliminary
+Given a graph 푋 and an associated Hermitian matrix 퐻, the transition matrix of its continuous-time quantum
+walk is
+푈(푡) = 푒−i푡퐻.
+We call 푋 a Hermitian graph if we do not assume any additional condition on the entries of 퐻. For the
+special case where 푋 is an oriented graph, we use the Hermitian matrix 퐻 defined as
+퐻푎,푏 =
+⎧
+⎪
+⎨
+⎪⎩
+i
+if there is an arc from 푎 to 푏 in 푋,
+−i
+if there is an arc from 푏 to 푎 in 푋, and
+0
+if there is no arc between 푎 and 푏 in 푋.
+Let 휃1, … , 휃푑 be distinct eigenvalues of 퐻. For 푟 = 1, … , 푑, let 퐸푟 denote the orthogonal projection
+matrix onto the 휃푟-eigenspace of 퐻. Then 퐸푟퐸푠 = 훿푟,푠퐸푟 and ∑
+푟 퐸푟 = 퐼. The spectral decomposition
+퐻 = ∑
+푟 휃푟퐸푟 gives
+푈(푡) =
+푑
+∑
+푟=1
+푒−i푡휃푟퐸푟.
+Given a unit vector 푣 ∈ ℂ푛, the system with initial state 푣 evolves to 푈(푡)푣 = ∑
+푟 푒−푖푡휃푟퐸푟푣 at time 푡. Therefore
+the pair (휃푟, 퐸푟) with 퐸푟푣 = 0 does not influence the state. We define the eigenvalue support of the vector
+푣 to be Φ푣 = {휃푟 ∶ 퐸푟푣 ≠ 0}. In the case 푣 = 푒푎 for some vertex 푎, we also call Φ푒푎 (Φ푎 for short) the
+eigenvalue support of 푎.
+Perfect state transfer from vertex 푎 to vertex 푏 occurs at time 휏 if
+푈(휏)푒푎 = 훼푒푏,
+(1)
+for some phase factor 훼. If 푎 = 푏 then we say the quantum walk is periodic at 푎.
+Multiplying 퐸푟 to both sides of Equation (1) gives
+푒−i휏휃푟퐸푟푒푎 = 훼퐸푟푒푏.
+(2)
+Hence, for 푟 = 1, … , 푑, there exists 푞푟(푎, 푏) ∈ [0, 2휋) such that
+퐸푟푒푎 = 푒i푞푟(푎,푏)퐸푟푒푏.
+(3)
+We say the vertices 푎 and 푏 are strongly cospectral when this condition is satisfied, and call 푞푟(푎, 푏) the quarrel
+from 푎 to 푏 relative to the eigenvalue 휃푟. Note that strongly cospectral vertices have the same eigenvalue
+support.
+We study perfect state transfer in oriented graphs and in Hermitian graphs in Sections 3 and 4. We give
+here a characterization of perfect state transfer in Hermitian graphs.
+Theorem 2.1. Perfect state transfer occurs from 푎 to 푏 in a Hermitian graph 푋 if and only if
+i. 푎 and 푏 are strongly cospectral vertices with quarrels 푞푟(푎, 푏), for 휃푟 ∈ Φ푎, and
+ii. for 휃푟, 휃푠, 휃ℎ, 휃퓁 ∈ Φ푎 such that ℎ ≠ 퓁, there exist integers 푚푟,푠 and 푚ℎ,퓁 satisfying
+휃푟 − 휃푠
+휃ℎ − 휃퓁
+=
+푞푟(푎, 푏) − 푞푠(푎, 푏) + 2푚푟,푠휋
+푞ℎ(푎, 푏) − 푞퓁(푎, 푏) + 2푚ℎ,퓁휋 .
+3
+
+Proof. From Equation (3), we see that perfect state transfer from 푎 to 푏 implies they are strongly cospectral.
+Suppose 푎 and 푏 are strongly cospectral with quarrel 푞푟(푎, 푏), for 휃푟 ∈ Φ푎(= Φ푏). Then Equation (1)
+holds if and only if for 휃푟, 휃푠 ∈ Φ푎,
+훼 = 푒i(푞푟(푎,푏)−휏휃푟) = 푒i(푞푠(푎,푏)−휏휃푠).
+(4)
+This is equivalent to
+푒i휏(휃푟−휃푠) = 푒i(푞푟(푎,푏)−푞푠(푎,푏))
+and
+휏
+(
+휃푟 − 휃푠
+)
+= 푞푟(푎, 푏) − 푞푠(푎, 푏) + 2푚푟,푠휋,
+for some integer 푚푟,푠. Condition (ii) follows immediately.
+We say the ratio condition on Φ푎 holds if
+휃푟 − 휃푠
+휃ℎ − 휃퓁
+∈ Q
+(5)
+for 휃푟, 휃푠, 휃ℎ, 휃퓁 ∈ Φ푎 such that ℎ ≠ 퓁.
+Theorem 2.2. In a Hermitian graph 푋, 푎 is periodic if and only if the ratio condition on Φ푎 holds.
+Proof. Note that 푞푟(푎, 푎) = 0 for 휃푟 ∈ Φ푎. The result follows immediately from Theorem 2.1.
+In Section 5, we consider a relaxation of perfect state transfer. A graph has pretty good state transfer
+from 푎 to 푏 if, for any 휀 > 0, there is a time 휏 satisfying
+|푈(휏)푎,푏| ≥ 1 − 휀.
+(6)
+Using the proof of Lemma 13.1 in [13], we conclude that if there is pretty good state transfer from 푎 to 푏
+then 푎 and 푏 are strongly cospectral. From
+푈(푡)푎,푏 =
+푑
+∑
+푟=1
+푒−i푡휃푟푒푇
+푎 퐸푟푒푏 =
+푑
+∑
+푟=1
+푒i(푞푟(푎,푏)−푡휃푟)(퐸푟)푏,푏,
+we see that there is pretty good state transfer from 푎 to 푏 if and only if for any 휖 > 0, there exists 휏 > 0 and
+훿휖 ∈ R such that
+|휏휃푟 − 푞푟(푎, 푏) − 훿휖| < 휖
+(mod 2휋),
+for 푟 ∈ Φ푎.
+Theorem 2.3. (Kronecker [14]) Let 휃1, … , 휃푑 and 푞1, … , 푞푑 be arbitrary real numbers. For any 휖 > 0, the
+system of inequalities
+|휃푟휏 − 푞푟| < 휖
+(mod 2휋),
+푟 = 1, … , 푑
+admits a solution for 휏 if and only if, for all set of integers 푙1, … , 푙푑,
+푙1휃1 + … + 푙푑휃푑 = 0
+implies
+푙1푞1 + … + 푙푑푞푑 = 0
+(mod 2휋).
+4
+
+Theorem 2.4. Let 푋 be Hermitian graph with eigenvalues 휃1, … , 휃푑 ∈ Φ푎. Then 푋 has pretty good state
+transfer from 푎 to 푏 if and only if the following conditions hold.
+i. The vertices 푎 and 푏 are strongly cospectral with quarrels 푞푟(푎, 푏), for 푟 = 1, … , 푑.
+ii. There exists 훿 ∈ R such that, for all integers 푙1, … , 푙푑 satisfying ∑푑
+푟=1 푙푟휃푟 = 0, we have
+푑
+∑
+푟=1
+푙푟
+(
+푞푟(푎, 푏) + 훿
+)
+= 0
+(mod 2휋).
+(7)
+Proof. The result follows from Proposition 4.01 of [15] and Theorem 2.3.
+Let 푆 be a set of vertices in 푋, we say multiple pretty good state transfer occurs on 푆 if there is pretty
+good state transfer between any two vertices in 푆. Section 5 gives two families of Hermitian graphs that have
+multiple pretty good state transfer.
+3
+Perfect state transfer in oriented graphs
+For graphs with real symmetric adjacency matrix, Kay shows that perfect state transfer cannot happen from
+one vertex to two distinct vertices [9]. This monogamous behaviour does not hold in Hermitian graphs with
+non-real entries. A graph has multiple perfect state transfer on a set 푆 of at least three vertices if there is
+perfect state transfer between any two vertices in 푆. When 푆 = 푉 (푋), we say 푋 has universal perfect
+state transfer. Lemma 22 of [10] gives a construction of Hermitian circulants that admit universal perfect
+state transfer. The oriented 3-cycle is a special case of this construction. In the same paper, Cameron et
+al. conjecture that the oriented 퐾2 and the oriented 퐾3 are the only oriented graphs that can have universal
+perfect state transfer. We confirm this conjecture in Section 3.1.
+In [11], Godsil and Lato investigated multiple perfect state transfer in oriented graph where 푆 is a proper
+subset of 푉 (푋). They give an example of an oriented graph on eight vertices that admits multiple perfect
+state transfer on a set of four vertices. In Section 3.2, we extend their example to an infinite family of oriented
+graphs that have multiple perfect state transfer.
+3.1
+Universal perfect state transfer
+In [1], Cameron et al. show that the oriented 퐾2 and 퐾3 with any orientation admit universal perfect state
+transfer. They give the following necessary conditions on the Hermitian graphs admitting universal perfect
+state transfer.
+Theorem 3.1. Let 퐻 be the matrix associated with a Hermitian graph 푋 that admits universal perfect state
+transfer. Then the following holds:
+1. All eigenvalues of 퐻 are simple.
+2. If 푃 is a unitary matrix diagonalizing 퐻 then |푃푎,푏| =
+1
+√
+푛, for 푎, 푏 ∈ 푉 (푋).
+3. Every vertex in 푋 is periodic.
+5
+
+Suppose 푋 is an oriented graph on 푛 vertices that has universal perfect state transfer. Let 퐻 be its
+associated Hermitian matrix with spectral decomposition
+퐻 =
+푛
+∑
+푟=1
+휃푟퐸푟.
+Then 퐸푟 has rank one with constant diagonal entries 푛−1. We see that 퐻2 has constant diagonal entries and
+the underlying (undirected) graph of 푋 is regular. Further, it follows from Theorem 6.1 of [11] that there
+exists a positive square-free integer Δ such that 휃푟 ∈ Z
+√
+Δ, for 푟 = 1, … , 푛. Hence
+min
+푟≠푠 |휃푟 − 휃푠| ≥
+√
+Δ.
+(8)
+We show in the following lemmas that an oriented graph with universal perfect state transfer can have at
+most eleven vertices.
+Lemma 3.2. Let 퐻 be a Hermitian matrix of order 푛 with zero diagonal entries. Let 휃1 ≤ 휃2 ≤ ⋯ ≤ 휃푛 be
+the eigenvalues of 퐻. Then
+푛
+∑
+푟,푠=1
+(
+휃푟 − 휃푠
+)2 = 2푛 Tr(퐻2).
+Proof. Observe that 휃푟 − 휃푠 is an eigenvalue of (퐻 ⊗ 퐼푛 − 퐼푛 ⊗ 퐻), for 푟, 푠 = 1 … , 푛. Hence
+푛
+∑
+푟,푠=1
+(
+휃푟 − 휃푠
+)2 = Tr
+(
+퐻 ⊗ 퐼푛 − 퐼푛 ⊗ 퐻
+)2 = Tr
+(
+퐻2 ⊗ 퐼푛 + 퐼푛 ⊗ 퐻2 − 2퐻 ⊗ 퐻
+)
+.
+The result follows from Tr(퐻 ⊗ 퐻) = 0.
+Lemma 3.3. Let 푋 be an oriented graph on 푛 vertices and 푚 edges with eigenvalues 휃1 < ⋯ < 휃푛. Let
+휎 = min푟≠푠 |휃푟 − 휃푠|. Then
+휎2 푛(푛2 − 1)
+24
+≤ 푚
+and
+휎2 ≤
+12
+푛 + 1.
+Proof. It follows from the definition of 휎 that 휎|푟 − 푠| ≤ |휃푟 − 휃푠|, and
+휎2
+푛
+∑
+푟,푠=1
+(푟 − 푠)2 ≤
+푛
+∑
+푟,푠=1
+(휃푟 − 휃푠
+)2 .
+The lower bound is
+휎2
+푛
+∑
+푟,푠=1
+(푟 − 푠)2 = 휎2
+⎛
+⎜
+⎜⎝
+2푛
+푛
+∑
+푟=1
+푟2 − 2
+( 푛
+∑
+푟=1
+푟
+)2⎞
+⎟
+⎟⎠
+= 휎2푛2(푛2 − 1)
+6
+.
+Applying Lemma 3.2 gives
+휎2 푛2(푛2 − 1)
+6
+≤ 2푛 Tr(퐻2) = 4푚푛.
+The second inequality in the lemma follows immediately from 푚 ≤
+(푛
+2
+)
+.
+6
+
+Corollary 3.4. Let 푋 be an oriented graph on 푛 vertices. If 푋 admits universal perfect state transfer then
+푛 ≤ 11. Further, if 푛 ≥ 6 then 푋 has integral eigenvalues.
+Proof. It follows from Equation (8) that 휎2 ≥ Δ ≥ 1. The second inequality of Lemma 3.3 gives 푛 ≤ 11.
+When 푛 ≥ 6, we have 휎2 < 2 which implies Δ = 1 and the eigenvalues of 푋 are integers.
+We are ready to rule out universal perfect state transfer in oriented graphs on more than three vertices.
+Theorem 3.5. The oriented 퐾2 and 퐾3 are the only oriented graphs admitting universal perfect state transfer.
+Proof. Suppose 푋 is an oriented graph on 푛 vertices that admits universal perfect state transfer. Then the
+underlying graph of 푋 is 푘-regular, for some integer 푘.
+Let 휃1 < ⋯ < 휃푛 be the eigenvalues of the Hermitian matrix 퐻 associated with 푋. Then 휃푟 ∈ Z
+√
+Δ,
+for some positive square-free integer Δ. Since i퐻 is a skew-symmetric matrix with entries ±1, we have
+휃푟 = −휃푛+1−푟
+for 푟 = 1, … , 푛.
+(9)
+Further, the characteristic polynomial of i퐻 is equal to the characteristic polynomial of its underlying graph
+over Z2.
+When 푛 = 4 or 5, 퐶푛 and 퐾푛 are the only regular graphs on 푛 vertices. An exhaustive search rules out
+oriented graphs on 4 or 5 vertices with spectrum satisying the above conditions.
+For 푛 ≥ 6, it follows from Lemma 3.3 and Corollary 3.4 that 휎 = min푟≠푠 |휃푟 − 휃푠| = 1 and
+푛2 − 1
+12
+≤ 푘 ≤ 푛 − 1.
+Using this inequality together with the fact that 푘 is even when 푛 is odd, we narrow down to the following
+possibilities.
+푛
+6
+7
+8
+9
+10
+11
+푘
+3, 4, 5
+4, 6
+6, 7
+8
+9
+10
+Applying Equation (9) to Tr(퐻2) yields
+푛푘 = 2
+⌊ 푛+1
+2 ⌋
+∑
+푟=1
+휃2
+푟 .
+Direct computation returns integral solutions to this equation for only three cases:
+푛
+푘
+underlying graph
+Possible spectrum of i퐻
+11
+10
+퐾11
+0, ±i, ±2i, ±3i, ±4i, ±5i
+7
+6
+퐾7
+0, ±i, ±2i, ±4i
+7
+4
+퐶7
+0, ±i, ±2i, ±3i
+It is straightforward to check that for each case, the characteristic polynomial of the underlying graph is not
+equal to the polynomial with the roots listed in the table over Z2.
+We conclude that there is no oriented graph on 푛 ≥ 4 vertices admitting universal perfect state transfer.
+7
+
+3.2
+Multiple perfect state transfer
+In [11], Godsil and Lato relax the notion of universal perfect state transfer to multiple perfect state transfer
+on a subset of vertices in oriented graphs. Let
+퐻⃖⃗퐶4 =
+⎡
+⎢
+⎢
+⎢⎣
+0
+−i
+0
+i
+i
+0
+−i
+0
+0
+i
+0
+−i
+−i
+0
+i
+0
+⎤
+⎥
+⎥
+⎥⎦
+be the Hermitian matrix of the directed 4-cycle. They show that the oriented graph with Hermitian matrix
+[1
+0
+0
+1
+]
+⊗ 퐻⃖⃗퐶4 +
+[ 0
+i
+−i
+0
+]
+⊗ 퐽4
+has multiple perfect state transfer on a set of four vertices.
+Making use of this technical lemma from [16], we extend the above example to an infinite family of
+oriented graphs where multiple perfect state transfer occur.
+Lemma 3.6. Let 퐴 and 퐵 be Hermitian matrices where 퐴 has spectral decomposition 퐴 = ∑
+푟 휃푟퐸푟. Then
+푒−푖푡(퐴⊗퐵) =
+∑
+푟
+퐸푟 ⊗ 푒−푖푡휃푟퐵.
+Lemma 3.7. Suppose 푋 is an oriented graph on 푛 vertices with associated Hermitian matrix 퐻푋, whose
+eigenvalues are odd integers. Let 푌 be the oriented graph with Hermitian matrix
+퐻푌 = 퐼푛 ⊗ 퐻⃖⃗퐶4 + 퐻푋 ⊗ 퐽4.
+Then �� admits multiple perfect state transfer on the set {4ℎ+1, 4ℎ+2, 4ℎ+3, 4ℎ+4}, for ℎ = 0, 1, … , 푛−1.
+Proof. Let 퐻푋 = ∑
+푟 휃푟퐸푟 be the spectral decomposition of 퐻푋. Since 퐼푛 ⊗ 퐻⃖⃗퐶4 and 퐻푋 ⊗ 퐽4 commute,
+applying Lemma 3.6 gives
+푒−i푡퐻푌 =
+(
+퐼푛 ⊗ 푒
+−i푡퐻⃗퐶4
+) (
+∑
+푟
+퐸푟 ⊗ 푒−i푡휃푟퐽4
+)
+=
+∑
+푟
+퐸푟 ⊗ 푒
+−i푡
+(
+퐻⃗퐶4
++휃푟퐽4
+)
+.
+For odd integer 휃푟, we have
+푒
+−i 휋
+4
+(
+퐻⃗퐶4
++휃푟퐽4
+)
+=
+⎡
+⎢
+⎢
+⎢⎣
+0
+−1
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+−1
+−1
+0
+0
+0
+⎤
+⎥
+⎥
+⎥⎦
+.
+Hence
+푒−i 휋
+4 퐻푌 = 퐼푛 ⊗
+⎡
+⎢
+⎢
+⎢⎣
+0
+−1
+0
+0
+0
+0
+−1
+0
+0
+0
+0
+−1
+−1
+0
+0
+0
+⎤
+⎥
+⎥
+⎥⎦
+,
+and, for ℎ = 0, 1, … , 푛 − 1, the vertex 4ℎ + 1 has perfect state transfer to 4ℎ + 4, 4ℎ + 3 and 4ℎ + 2 at time
+휋
+4, 휋
+2 and 3휋
+4 , respectively.
+8
+
+If 푋 is obtained by orienting all edges in the (2푚 + 1)-cube from one bipartition to the other bipartition,
+then its associated matrix has the form
+퐻푋 =
+[
+0
+i퐵
+−i퐵푇
+0
+]
+.
+Then 퐻푋 has the same spectrum as the adjacency matrix of the (undirected) (2푚 + 1)-cube, which consists
+of only odd integers. Lemma 3.7 gives an oriented graph admitting multiple perfect state transfer for integer
+푚 ≥ 0. When 푚 = 0, then 푌 is the oriented graph given in [11].
+4
+Perfect state transfer in Hermitian graphs
+We focus on Hermitian graphs with algebraic entries in the first part of this section. In particular, we study
+the phase factors when perfect state transfer occurs in these graphs in Section 4.1.
+Suppose 푋 is a Hermitian graph with algebraic entries. By Theorem 6.1 of [2] and Theorem 2.2, if
+perfect state transfer from 푎 to 푏 occurs then the quantum walk on 푋 is periodic at both 푎 and 푏. Section 4.2
+gives examples of Hermitian graphs (with transcendental entries) in which perfect state transfer occurs from
+푎 to 푏 but 푎 and 푏 are not periodic.
+4.1
+Phase factor
+We restrict our attention to Hermitian graphs with algebraic entries and extract information about the phase
+factor when perfect state transfer occurs.
+Let 퐻 be an algebraic Hermitian matrix. Its characteristic polynomial has algebraic coefficients. Given
+spectral decomposition 퐻 = ∑
+푟 휃푟퐸푟, the eigenvalues 휃푟’s are algebraic so are the entries in 퐸푟.
+Theorem 4.1. Let 퐻 be an algebraic matrix associated with a Hermitian graph with spectral decomposition
+퐻 = ∑
+푟 휃푟퐸푟. If perfect state transfer occurs from 푎 to 푏 with phase factor 훼, then 훼 is algebraic if and only
+if
+휃푟
+휃푠
+∈ Q,
+for 휃푟, 휃푠 ∈ Φ푎 such that 휃푠 ≠ 0.
+Proof. Suppose perfect state transfer occurs from 푎 to 푏 at time 휏 with algebraic phase factor 훼. It follows
+from Equation (2) that 푒−i휏휃푟 is algebraic, for 휃푟 ∈ Φ푎 = Φ푏. Applying the Gelfond-Schneider Theorem to
+(푒−i휏휃푠) 휃푟
+휃푠 = 푒−i휏휃푟,
+for 휃푟, 휃푠 ∈ Φ푎 with 휃푠 ≠ 0, we conclude that 휃푟
+휃푠 is rational.
+Now suppose 휃푠
+휃푟 ∈ Q for 휃푟, 휃푠 ∈ Φ푎 with 휃푠 ≠ 0. Let 푞푟(푎, 푏) be the quarrels from 푎 to 푏 relative to
+휃푟 ∈ Φ푎. It follows from Equation (3) that 푒i푞푟(푎,푏) is algebraic. Applying Equation (4) yields
+훼
+( 휃푟
+휃푠 −1
+)
+=
+(
+푒i(푞푠(푎,푏)−휏휃푠)) 휃푟
+휃푠 푒i(휏휃푟−푞푟(푎,푏)) =
+(
+푒i푞푠(푎,푏)) 휃푟
+휃푠 푒−i푞푟(푎,푏).
+The right-hand side is algebraic, so is 훼.
+9
+
+Theorem 4.2. Let 퐻 be an algebraic matrix associated with a Hermitian graph with spectral decomposition
+퐻 = ∑
+푟 휃푟퐸푟. Suppose perfect state transfer occurs from 푎 to 푏 with phase factor 훼. If there exist integers
+푘푟’s satisfying
+∑
+푟∈Φ푎
+푘푟휃푟 = 0
+and
+∑
+푟∈Φ푎
+푘푟 ≠ 0
+then 훼 is algebraic.
+Proof. From Equation (4), we have
+훼
+∑
+푟∈Φ푎 푘푟 = 푒
+−i휏
+(∑
+푟∈Φ푎 푘푟휃푟
+) ∏
+푟∈Φ푎
+(푒i푞푟(푎,푏))푘푟 =
+∏
+푟∈Φ푎
+(푒i푞푟(푎,푏))푘푟 .
+Since the right-hand side is algebraic and ∑
+푟∈Φ푎 푘푟 ≠ 0, we conclude that 훼 is algebraic.
+We apply the theorem to algebraic Hermitian graphs where Φ푎 contains all eigenvalues of 퐻.
+Corollary 4.3. Let 퐻 be an algebraic matrix associated with a Hermitian graph with zero diagonal entries.
+Suppose perfect state transfer occurs from 푎 to 푏 with phase factor 훼. If 푎 has full eigenvalue support then 훼
+is algebraic.
+Proof. Let 푘푟 be the multiplicity of 휃푟, for 휃푟 ∈ Φ푎. Since Φ푎 contains all eigenvalues of 퐻, we have
+∑
+푟∈Φ푎 푘푟휃푟 = Tr(퐻) = 0 and ∑
+푟∈Φ푎 푘푟 equals the number of vertices. It follows from Theorem 4.2 that the
+phase factor at perfect state transfer is algebraic.
+Given spectral decomposition of an algebraic Hermitian matrix 퐻 = ∑
+푟 휃푟퐸푟, if 퐸푟 has constant diagonal
+then every vertex has full eigenvalue support. In particular, Corollary 4.3 applies to
+• the adjacency matrix of a walk regular graph,
+• an algebraic Hermitian matrix with zero diagonal that belongs to a Bose-Mesner algebra, and
+• Hermitian circulants with algebraic entries and zero diagonal.
+4.2
+One-way perfect state transfer
+We saw at the beginning of Section 4 that if perfect state transfer occurs from 푎 to 푏 in an algebraic Hermitian
+graph then both 푎 and 푏 are periodic. In particular, there is perfect state transfer from 푏 back to 푎.
+We give a family of Hermitian graphs, with transcendental entries, that have perfect state transfer from
+푎 to 푏 but not periodic at 푎 nor 푏. In particular, they do not have perfect state transfer from 푏 to 푎.
+Theorem 4.4. There exist infintely many Hermitian graphs which admit perfect state transfer from 푎 to 푏 but
+are not periodic at 푎.
+Proof. Let 휆 be any real number such that 휆 ∉ Q휋. Define matrices
+푃 = 1
+2
+⎡
+⎢
+⎢
+⎢⎣
+1
+1
+1
+1
+1
+1
+−1
+−1
+1
+−1
+푒i휆
+−푒i휆
+1
+−1
+−푒i휆
+푒i휆
+⎤
+⎥
+⎥
+⎥⎦
+and
+퐷 =
+⎡
+⎢
+⎢
+⎢⎣
+0
+0
+0
+0
+0
+휋
+0
+0
+0
+0
+휆
+0
+0
+0
+0
+휆 + 휋
+⎤
+⎥
+⎥
+⎥⎦
+.
+10
+
+Consider the Hermitian matrix
+퐻 ∶= 푃 퐷푃 −1 =
+(휋 + 휆
+2
+)
+퐼4 −
+⎡
+⎢
+⎢
+⎢
+⎢⎣
+0
+휆
+2
+휋
+4(1 + 푒−푖휆)
+휋
+4(1 − 푒−푖휆)
+휆
+2
+0
+휋
+4(1 − 푒−푖휆)
+휋
+4(1 + 푒−푖휆)
+휋
+4 (1 + 푒푖휆)
+휋
+4(1 − 푒푖휆)
+0
+휆
+2
+휋
+4 (1 − 푒푖휆)
+휋
+4(1 + 푒푖휆)
+휆
+2
+0
+⎤
+⎥
+⎥
+⎥
+⎥⎦
+.
+Let 휃1 = 0, 휃2 = 휋, 휃3 = 휆 and 휃4 = 휆 + 휋. All vertices have full eigenvalue support. Vertices 1 and
+3 are strongly cospectral with quarrels: 푞1(3, 1) = 0, 푞2(3, 1) = 휋, 푞3(3, 1) = 휆, and 푞4(3, 1) = 휆 + 휋. By
+Theorem 2.1, we have perfect state transfer from vertex 3 to 1 at time 휏 = 1 with phase factor 1. As 휆 is not
+a rational multiple of 휋, we have
+휃3 − 휃1
+휃2 − 휃1
+= 휆
+휋 ∉ ℚ.
+By Theorem 2.2, 퐻 is not periodic at vertex 1 nor at vertex 3.
+Example 4.5. Consider the complex Hadamard matrix
+푃 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+1
+1
+1
+푖
+푖
+푖
+푖
+1
+−1
+푒푖휃
+−푒푖휃
+−1
+1
+−푒푖휃
+푒푖휃
+1
+1
+푒푖2휃
+푒푖2휃
+−푖
+−푖
+−푖푒푖2휃
+−푖푒푖2휃
+1
+−1
+푒푖3휃
+−푒푖3휃
+1
+−1
+푒푖3휃
+−푒푖3휃
+푖
+푖
+−푖
+−푖
+−1
+−1
+1
+1
+−푖
+푖
+푖푒푖휃
+−푖푒푖휃
+푖
+−푖
+−푖푒푖휃
+푖푒푖휃
+푖
+푖
+−푖푒푖2휃
+−푖푒푖2휃
+1
+1
+−푒푖2휃
+−푒푖2휃
+−푖
+푖
+푖푒푖3휃
+−푖푒푖3휃
+−푖
+푖
+푖푒푖3휃
+−푖푒푖3휃
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+and diagonal matrix 퐷 = diag
+(
+0, 휋, 휃, 휃 + 휋, 휋
+2, 3휋
+2 , 휃 + 휋
+2, 휃 + 3휋
+2
+)
+. Then the Hermitian graph 푋 with
+matrix 퐻 = 푃 퐷푃 −1 admit perfect state transfer from vertex 1 to 2 at 푡 = 1, from vertex 1 to 3 at 푡 = 2, from
+vertex 1 to 4 at 푡 = 3. Each vertex has full eigenvalue support, and if 휃 ∉ Q휋, then the ratio condition is not
+satisfied and 푋 is not periodic at any vertex.
+5
+Multiple pretty good state transfer
+Theorem 4.4 shows that it is possible to have one-way perfect state transfer in Hermitian graph. We now
+show that pretty good state transfer in Hermitian graphs goes both ways.
+Lemma 5.1. If a Hermitian graph admits pretty good state transfer from 푎 to 푏, then it has pretty good state
+transfer from 푏 to 푎.
+Proof. Suppose 푈(푡) is the transition matrix of a Hermitian graph that has pretty good state transfer from
+푎 to 푏. Then, for 휀 > 0, there exists a time 휏1 such that 푈(휏1)푒푎 = 훾1푒푏 + 휌1, for some phase factor 훾1 and
+vector 휌1 with ‖휌1‖ < 휀
+2.
+As 푈(푡) is almost periodic, there exists 휏2 > 휏1 such that 푈(휏2)푒푎 = 훾2푒푎 + 휌2, for some phase factor 훾2
+and some vector 휌2 with ‖휌2‖ < 휀
+2. We have
+푈(휏2 − 휏1)푒푏 = 훾1푈(휏2)
+(
+푒푎 − 푈(−휏1)휌1
+)
+= 훾1
+(
+훾2푒푎 + 휌2 − 푈(휏2 − 휏1)휌1
+)
+.
+11
+
+Hence
+‖푈(휏2 − 휏1)푒푏 − 훾1훾2푒푎‖ = ‖휌2 − 푈(휏2 − 휏1)휌1‖ ≤ ‖휌1‖ + ‖휌2‖ < 휀
+and there is pretty good state transfer from 푏 to 푎.
+In [5], Zimborás et al. assign a complex weight 푒i훽 to an edge in the following graph and use the weight
+to control the fidelity at 푏 and 푐 with initial state 푒푎.
+푒i훽
+푎
+푏
+푐
+This graph can be viewed as the rooted product of the weighted 퐾3 with a path. Given a graph 푋 on 푛 vertices
+and a rooted graph 푌 with root 푎. The rooted product of 푋 and 푌 , 푋◦푌 , is obtained by taking 푛 isomorphic
+copies of 푌 and identifying the 푗-th vertex of 푋 with the root of the 푗-th copy of 푌 . In this section, we give
+two families of rooted products that have multiple pretty good state transfer.
+5.1
+Oriented 3-cycle rooted with a star
+In [3], Fan and Godsil show that the double star, the rooted product of 퐾2 and 퐾1,푚, has pretty good state
+transfer between the two non-pendant vertices if and only if 4푚 + 1 is not a perfect square. Note that 퐾2 is
+the only simple undirected graph with universal perfect state transfer. We extend their result to the rooted
+product of the oriented 3-cycle ⃖⃖⃗
+퐾3 with ̂
+퐾1,푚, where ̂
+퐾1,푚 denotes the star 퐾1,푚 with the non-pendant vertex
+being its root.
+푐
+푎
+푏
+Lemma 5.2. Suppose 푎 and 푏 are strongly cospectral vertices in the Hermitian graph 푋 on 푛 ≥ 2 vertices.
+Then they are strongly cospectral in the rooted product 푋◦ ̂
+퐾1,푚.
+Proof. Let 퐻푋 be the Hermitian matrix associated with 푋 with spectral decomposition 퐻푋 = ∑푑
+푟=1 휃푟퐸푟 .
+Then the matrix associated with the rooted product 푌 = 푋◦ ̂
+퐾1,푚 is
+퐻푌 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+0
+0
+⋯
+0
+0
+0
+0
+⋯
+0
+⋮
+⋮
+⋮
+⋱
+⋮
+0
+0
+0
+⋯
+0
+0
+0
+0
+⋯
+0
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+⊗ 퐻푋 +
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+0
+1
+1
+⋯
+1
+1
+0
+0
+⋯
+0
+⋮
+⋮
+⋮
+⋱
+⋮
+1
+0
+0
+⋯
+0
+1
+0
+0
+⋯
+0
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+⊗ 퐼푛.
+12
+
+For 푟 = 1, … , 푑, define
+휆±
+푟 =
+휃푟 ±
+√
+휃2
+푟 + 4푚
+2
+,
+and
+퐹 ±
+푟 =
+1
+(휆±
+푟 )2 + 푚
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+(휆±
+푟 )2
+휆±
+푟
+휆±
+푟
+⋯
+휆±
+푟
+휆±
+푟
+1
+1
+⋯
+1
+⋮
+⋮
+⋮
+⋱
+⋮
+휆±
+푟
+1
+1
+⋯
+1
+휆±
+푟
+1
+1
+⋯
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+⊗ 퐸푟.
+Define
+퐹0 =
+⎡
+⎢
+⎢⎣
+0
+ퟎ푚
+ퟎ푚
+퐼푚 − 1
+푚퐽푚
+⎤
+⎥
+⎥⎦
+⊗ 퐼푛.
+Then 퐻푌 has spectral decomposition
+퐻푌 = 0 ⋅ 퐹0 +
+푑
+∑
+푟=1
+(휆+
+푟 ⋅ 퐹 +
+푟 + 휆−
+푟 ⋅ 퐹 −
+푟
+) .
+(10)
+Note that the (1, 1)-block are indexed by the vertices in 푋 and the eigenvalue 0 is not in the support of 푎 nor
+푏. The result follows from the (1, 1)-block of 퐹 +
+푟 and 퐹 −
+푟 being non-zero scalar multiple of 퐸푟.
+Corollary 5.3. Suppose 푋 is a Hermitian graph with universal perfect state transfer with spectrum Φ. Let
+푆 be the set of non-pendant vertices in 푋◦ ̂
+퐾1,푚. Let
+Ψ =
+{
+휃 ±
+√
+휃2 + 4푚
+2
+||| 휃 ∈ Φ
+}
+.
+If Ψ is linearly independent over Q, then 푋◦ ̂
+퐾1,푚 has multiple pretty good state transfer on 푆.
+Proof. For 푎, 푏 ∈ 푆, there is perfect state transfer between 푎 and 푏 in 푋, so 푎 and 푏 are strongly cospectral in
+푋◦ ̂
+퐾1,푚 by Lemma 5.2. We see in Equation (10) that Ψ is the eigenvalue support of 푎 in the rooted product.
+It follows from Theorem 2.4 that pretty good state transfer occurs between 푎 and 푏 in 푋◦ ̂
+퐾1,푚.
+In the following result, we focus on 푋 = ⃖⃖⃗
+퐾3 which has spectral decomposition
+⎡
+⎢
+⎢⎣
+0
+−i
+i
+i
+0
+−i
+−i
+i
+0
+⎤
+⎥
+⎥⎦
+= 0 ⋅ 1
+3퐽3 +
+√
+3 ⋅ 1
+3
+⎡
+⎢
+⎢⎣
+1
+푒−2휋i∕3
+푒2휋i∕3
+푒2휋i∕3
+1
+푒−2휋i∕3
+푒−2휋i∕3
+푒2휋i∕3
+1
+⎤
+⎥
+⎥⎦
+−
+√
+3 ⋅ 1
+3
+⎡
+⎢
+⎢⎣
+1
+푒2휋i∕3
+푒−2휋i∕3
+푒−2휋i∕3
+1
+푒2휋i∕3
+푒2휋i∕3
+푒−2휋i∕3
+1
+⎤
+⎥
+⎥⎦
+.
+Hence any two vertices in ⃖⃖⃗
+퐾3 are strongly cospectral. Let 푉 (⃖⃖⃗
+퐾3) = {푎, 푏, 푐}. Then the eigenvalue support
+of 푎 in ⃖⃖⃗
+퐾3◦ ̂
+퐾1,푚 are 휆1 =
+√
+푚, 휆2 = −
+√
+푚,
+휆3 =
+√
+3 +
+√
+3 + 4푚
+2
+,
+휆4 =
+√
+3 −
+√
+3 + 4푚
+2
+,
+휆5 = −
+√
+3 +
+√
+3 + 4푚
+2
+and
+휆6 = −
+√
+3 −
+√
+3 + 4푚
+2
+.
+13
+
+From Equation (10), the quarrels in ⃖⃖⃗
+퐾3◦ ̂
+퐾1,푚 are
+푞푟(푎, 푏) =
+⎧
+⎪
+⎨
+⎪⎩
+0
+if 푟 = 1, 2,
+2휋
+3
+if 푟 = 3, 4, and
+−2휋
+3
+if 푟 = 5, 6.
+Theorem 5.4. The rooted product ⃖⃖⃗
+퐾3◦ ̂
+퐾1,푚 admits multiple pretty good state transfer on the set {푎, 푏, 푐} of
+non-pendant vertices if and only if one of the following holds.
+1. gcd(3, 푚) = 1.
+2. 푚 = 3푠, for some integer 푠 such that neither 푠 nor 4푠 + 1 are perfect square.
+3. 푚 = 27푘2, for some integer 푘.
+4. 푚 = 27푘2 + 27푘 + 6, for some integer 푘.
+Proof. Since ⃖⃖⃗
+퐾3◦ ̂
+퐾1,푚 has an automorphism that maps 푎 to 푏, 푏 to 푐 and 푐 to 푎, it is sufficient to prove that
+there is pretty good state transfer from 푎 to 푏 in the rooted product.
+By Lemma 5.2, Condition (i) of Theorem 2.4 holds. For Condition (ii) of Theorem 2.4, we consider
+integers 푙1, … , 푙6 satisfying
+6
+∑
+푟=1
+푙푟휆푟 =
+(
+푙1 − 푙2
+) √
+푚 +
+(푙3 + 푙4 − 푙5 − 푙6
+2
+) √
+3 +
+(푙3 − 푙4 + 푙5 − 푙6
+2
+) √
+3 + 4푚 = 0.
+(11)
+Case 1: If gcd(3, 푚) = 1 then the set {
+√
+3,
+√
+푚,
+√
+3 + 4푚} is linearly independent over Q. Equation (11)
+implies (푙3 + 푙4 − 푙5 − 푙6)∕2 = 0 and
+6
+∑
+푟=1
+푙푟푞푟(푎, 푏) =
+(
+푙3 + 푙4 − 푙5 − 푙6
+) 2휋
+3 = 0
+(mod 2휋).
+(12)
+Condition (ii) of Theorem 2.4 holds with 훿 = 0, so there is pretty good state transfer from 푎 to 푏 in
+⃖⃖⃗
+퐾3◦ ̂
+퐾1,푚.
+Case 2: When 푚 = 3푠, Equation (11) becomes
+(
+푙1 − 푙2
+) √
+푠 +
+(푙3 + 푙4 − 푙5 − 푙6
+2
+)
++
+(푙3 − 푙4 + 푙5 − 푙6
+2
+) √
+1 + 4푠 = 0.
+If 푠 and 4푠 + 1 are not perfect squares then {1,
+√
+푠,
+√
+1 + 4푠} is linearly independent over Q and
+Equation (11) implies Equation (12). Hence there is pretty good state transfer from 푎 to 푏.
+Case 3: Suppose 푚 = 3ℎ2, for some integer ℎ. Then 4ℎ2 + 1 is not a perfect square, and Equation (11)
+becomes
+(2ℎ(푙1 − 푙2) + 푙3 + 푙4 − 푙5 − 푙6
+2
+)
++
+(푙3 − 푙4 + 푙5 − 푙6
+2
+) √
+4ℎ2 + 1 = 0,
+14
+
+which implies 푙3 + 푙4 − 푙5 − 푙6 = −2ℎ(푙1 − 푙2). If ℎ = 3푘, for some integer 푘, then Equation (12)
+holds and pretty good state transfer occurs from 푎 to 푏.
+Suppose ℎ is not divisible by 3. Equation (11) holds when 푙1 = 푙2 = 푙4 = 푙5 = 0 and 푙3 = 푙6 = 1.
+Since
+6
+∑
+푟=1
+푙푟
+(푞푟(푎, 푏) + 훿) = 2훿,
+Equation (7) holds if and only if 훿 ∈ Z휋.
+Equation (11) also holds when 푙1 = 1, 푙2 = 푙3 = 푙4 = 0, 푙5 = 푙6 = ℎ, but
+6
+∑
+푟=1
+푙푟(푞푟(푎, 푏) + 훿) = −4ℎ휋
+3
++ (2ℎ + 1)훿 ≠ 0
+(mod 2휋)
+when 훿 ∈ Z휋. We conclude that pretty good state transfer from 푎 to 푏 does not occur.
+Case 4: Suppose 푚 = 3푠 with 4푠 + 1 = ℎ2, for some integer ℎ. Then 푠 is not a perfect square, and Equa-
+tion (11) becomes
+(푙1 − 푙2)
+√
+푠 + (푙3 + 푙4 − 푙5 − 푙6) + ℎ(푙3 − 푙4 + 푙5 − 푙6)
+2
+= 0,
+which implies 푙3 + 푙4 − 푙5 − 푙6 = −ℎ(푙3 − 푙4 + 푙5 − 푙6). If ℎ is divisible by 3 then Equation (12)
+holds and pretty good state transfer occurs from 푎 to 푏. In this case, 푚 = 27푘2 + 27푘 + 6 if we write
+4푠 + 1 = 32(2푘 + 1)2.
+If ℎ is not divisible by 3, Equation (11) holds when 푙1 = 푙2 = 푙4 = 푙5 = 0, 푙3 = 푙6 = 1 and when
+푙1 = 푙2 = 0, 푙3 = 푙4 = ℎ, 푙5 = −1 and 푙6 = 1. Using the same argument as in the previous case, we
+see that there does not exist 훿 satisfying Equation (7) for both assignments for the 푙푗’s. We conclude
+that pretty good state transfer from 푎 to 푏 does not occur.
+5.2
+Circulants rooted with a looped path
+In [4], Kempton et al. show that a path with a loop on each end-vertex with transcendental weight 훾 has
+pretty good state transfer between the two end-vertices. We use 푃 훾
+푚 to denote the rooted path on vertices
+{1, 2, … , 푚} that has root 푚 and a loop on vertex 1 with weight 훾. Then the path of length 2푚 − 1 with a
+loop of weight 훾 on each end-vertex studied in [4] can be viewed as the rooted product of 퐾2 with 푃 훾
+푚.
+Path 푃 훾
+푚 rooted at 푚 with a loop at 1
+1
+2
+푚
+훾
+15
+
+We extend their result to the rooted product 푋◦푃 훾
+푚 where 푋 is Hermitian circulant with rational eigen-
+values that admits universal perfect state transfer. Orthogonal polynomials and field trace are the main tools
+used in this section. Please see Chapter 8 of [17] for the background of orthogonal polynomials, and see [4]
+and Chapter 14 of [18] for some basic facts on field trace.
+Suppose 푉 (푋) = {푥0, 푥1, … , 푥푛−1}. Then we label the vertices of 푋◦푃 훾
+푚 with the ordered pair (푥ℎ, 푗)
+denoting the 푗-th vertex on 푃 훾
+푚 that is rooted at 푥ℎ in 푋, for ℎ = 0, 1, … , 푛 − 1 and 푗 = 1, … , 푚.
+(푥0, 푚)
+(푥1, 푚)
+(푥2, 푚)
+(푥0, 1)
+(푥0, 2)
+훾
+(푥1, 1)
+(푥1, 2)
+훾
+(푥2, 1)
+(푥2, 2)
+훾
+The rooted product of ⃖⃖⃗
+퐾3 with 푃 훾
+푚
+Let 퐻푋 be the matrix of the Hermitian circulant 푋 with universal perfect state transfer. It follows from
+Theorem 8 of [1] that the eigenvalues of 퐻푋 are simple. Given distinct eigenvalues 휃0, 휃1, … , 휃푛−1 of 퐻푋
+and the discrete Fourier matrix of order 푛
+퐹푛 =
+1
+√
+푛
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+1
+1
+1
+⋯
+1
+1
+휁
+휁2
+⋯
+휁푛−1
+1
+휁2
+휁4
+⋯
+휁2(푛−1)
+⋮
+⋮
+⋮
+⋱
+⋮
+1
+휁푛−1
+휁2(푛−1)
+⋯
+휁(푛−1)2
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+where 휁 = 푒2휋i∕푛, we can write
+퐻푋 = 퐹푛
+⎡
+⎢
+⎢
+⎢⎣
+휃0
+0
+⋯
+0
+0
+휃1
+⋯
+0
+⋮
+⋮
+⋱
+⋮
+0
+0
+⋯
+휃푛−1
+⎤
+⎥
+⎥
+⎥⎦
+퐹 ∗
+푛 .
+For 0 ≤ 푎, 푏 ≤ 푛 − 1, the vertices 푥푎 and 푥푏 are strongly cospectral with quarrel
+푞푗(푥푎, 푥푏) = 2휋푗(푏 − 푎)
+푛
+,
+(13)
+for 푗 = 0, 1, … , 푛 − 1.
+Theorem 22 of [1] gives the following characterization of Hermitian circulants that have universal perfect
+state transfer.
+Theorem 5.5. Let 푋 be a Hermitian circulant on 푛 vertices with simple eigenvalues 휃0, … , 휃푛−1. Then 푋
+has universal perfect state transfer if and only if there exist 훼, 훽 ∈ R with 훽 > 0, 푐0, … , 푐푛−1 ∈ Z and integer
+ℎ coprime with 푛 such that
+휃푗 = 훼 + 훽 (푗ℎ + 푐푗푛) ,
+for 푗 = 0, … , 푛 − 1.
+16
+
+To determine the spectrum of 푍 = 푋◦푃 훾
+푚, we consider the 푚 × 푚 Jacobi matrices
+푇푗 ∶=
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+훾
+1
+0
+⋯
+0
+0
+1
+0
+1
+⋯
+0
+0
+0
+1
+0
+⋯
+0
+0
+⋮
+⋮
+⋮
+⋱
+⋮
+⋮
+0
+0
+0
+⋯
+0
+1
+0
+0
+0
+⋯
+1
+휃푗
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+,
+for 푗 = 0, 1, … , 푛 − 1.
+(14)
+Let 휑푗,0 = 1 and let 휑푗,푟(푡) be the characteristic polynomial of the 푟-th leading principal submatrix of 푇푗, for
+푟 = 1, … , 푚. Then 휑푗,0(푡), 휑푗,1(푡), … , 휑푗,푚(푡) is a sequence of orthogonal polynomials satisfying 휑푗,0(푡) = 1,
+휑푗,1(푡) = 푡 − 훾,
+휑푗,푟(푡) = 푡 휑푗,푟−1(푡) − 휑푗,푟−2(푡)
+(15)
+for 푟 = 2, … , 푚 − 1, and
+휑푗,푚(푡) = (푡 − 휃푗
+) 휑푗,푚−1(푡) − 휑푗,푚−2(푡).
+(16)
+From Lemma 8.5.2 of [17], the roots 휆푗,1, … , 휆푗,푚 of 휑푗,푚(푡) = 0 are the eigenvalues of 푇푗. Further,
+Φ푗,푠 =
+[1
+휑푗,1(휆푗,푠)
+…
+휑푗,푚−1(휆푗,푠)]푇
+is an eigenvector of 푇푗 corresponding to eigenvalue 휆푗,푠, for 푠 = 1, … , 푚. It follows from Lemma 8.1.1 of
+[17] that the eigenvalues of 푇푗 are simple. It is also known that consecutive orthogonal polynomials do not
+have non-trivial common factor.
+The Hermitian matrix of 푍 is
+퐻푍 =
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+0
+0
+⋯
+0
+0
+0
+0
+⋯
+0
+0
+⋮
+⋮
+⋱
+⋮
+⋮
+0
+0
+⋯
+0
+0
+0
+0
+⋯
+0
+1
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+⊗ 퐻푋 +
+⎡
+⎢
+⎢
+⎢
+⎢
+⎢⎣
+훾
+1
+⋯
+0
+0
+1
+0
+⋯
+0
+0
+⋮
+⋮
+⋱
+⋮
+⋮
+0
+0
+⋯
+0
+1
+0
+0
+⋯
+1
+0
+⎤
+⎥
+⎥
+⎥
+⎥
+⎥⎦
+⊗ 퐼푛.
+(17)
+Since 퐻푋퐹푛푒푗 = 휃푗퐹푛푒푗, we have
+퐻푍
+(Φ푗,푠 ⊗ 퐹푛푒푗
+) = 휆푗,푠
+(Φ푗,푠 ⊗ 퐹푛푒푗
+)
+(18)
+for 푗 = 0, … , 푛 − 1 and 푠 = 1, … , 푚.
+Lemma 5.6. Let 푋 be a Hermitian circulant with distinct eigenvalues 휃0, 휃1, … , 휃푛 and let 퐹푛, 휆푗,푠, and Φ푗,푠
+be defined as above. For 푗 = 0, … , 푛 − 1 and 푠 = 1, … , 푚, 휆푗,푠 is a simple eigenvalue of the Hermitian
+graph 푍 defined in Equation (17), with spectral decomposition
+퐻푍 =
+푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+휆푗,푠
+1
+‖Φ푗,푠‖2
+(
+Φ푗,푠Φ∗
+푗,푠
+)
+⊗
+(
+(퐹푛푒푗)(퐹푛푒푗)∗)
+.
+For 푥푎, 푥푏 ∈ 푉 (푋) and ℎ = 1, … , 푚, the vertices (푥푎, ℎ) and (푥푏, ℎ) are strongly cospectral in 푍 with
+quarrel corresponding to eigenvalues 휆푗,푠 being
+푞푗,푠
+(
+(푥푎, ℎ), (푥푏, ℎ)
+)
+= 2휋푗(푏 − 푎)
+푛
+,
+for 푗 = 0, … , 푛 − 1 and 푠 = 1, … , 푚.
+17
+
+Proof. It is sufficient to show that the eigenvalues 휆푗,푠 of 푍, for 푗 = 0, … , 푛−1 and 푠 = 1, … , 푚, are distinct.
+Supoose 휆푗1,푠1 = 휆푗2,푠2. From Equation (15), we have
+휑푗1,푟
+(휆푗1,푠1
+) = 휑푗2,푟
+(휆푗2,푠2
+) ,
+for 푟 = 1, … , 푚 − 1. From Equation (16), 휑푗1,푚
+(휆푗1,푠1
+) = 휑푗2,푚
+(휆푗2,푠2
+) = 0 implies 휃푗1 = 휃푗2 and 푗1 = 푗2.
+Since 휑푗1,푚(푡) = 0 has 푚 distinct roots, we conclude that 푠1 = 푠2.
+We get the quarrels of 푍 directly from Equations (18) and (13).
+For the rest of this section, we assume that 훾 is transcendental and 휃0, 휃1, … , 휃푛−1 ∈ Q as in Theorem 5.8.
+Applying Laplace expansion along the first two rows of 푇푗 in Equation (14) gives
+휑푗,푚(푡) = (푡 − 훾)푔푛−1(푡) − 푔푛−2(푡),
+where 푔푛−1(푡) is the characteristic polynomial of the (푛 − 1) × (푛 − 1) Jacobi matrix
+⎛
+⎜
+⎜
+⎜
+⎜
+⎜⎝
+휃푗
+1
+⋯
+0
+0
+1
+0
+⋯
+0
+0
+⋮
+⋮
+⋱
+⋮
+⋮
+0
+0
+⋯
+0
+1
+0
+0
+⋯
+1
+0
+⎞
+⎟
+⎟
+⎟
+⎟
+⎟⎠
+,
+and 푔푛−2(푡) is the characteristic polynomial of its (푛 − 2)-th leading principal submatrix. Now 푔푛−1(푡) and
+푔푛−2(푡) are consecutive orthogonal polynomials, so they do not have any common factor of positive degree.
+Since 푔푛−1(푡) and 푔푛−2(푡) are rational polynomials and 훾 is transcendental, we conclude that 휑푗,푚(푡) is irre-
+ducible over Q(훾). Then the splitting field 퐹푗 of 휑푗,푚(푡) is a Galois extension over Q(훾).
+Given a Galois extension 퐸∕퐾, we use Tr퐸∕퐾(휇) to denote the trace of 휇 from 퐸 to 퐾. Here are some
+properties of the trace map useful for the proof of Theorem 5.8.
+Theorem 5.7. Let 퐸∕퐾 be a Galois extension. The following properties hold.
+i. For 휇 ∈ 퐸, Tr퐸∕퐾(휇) ∈ 퐾.
+ii. For 휇 ∈ 퐾, Tr퐸∕퐾(휇) = [퐸 ∶ 퐾]휇.
+iii. For 휇1, 휇2 ∈ 퐸, Tr퐸∕퐾(휇1 + 휇2) = Tr퐸∕퐾(휇1) + Tr퐸∕퐾(휇2).
+iv. If 퐾 ⊂ 퐹 ⊂ 퐸 are extension fields, then Tr퐸∕퐾(휇) = Tr퐹∕퐾
+(
+Tr퐸∕퐹 (휇)
+)
+.
+v. If the minimal polynomial of 휇 ∈ 퐸 over 퐾 is 푡푚 + 푎푚−1푡푚−1 + ⋯ + 푐0 then
+Tr퐸∕퐾(휇) = −[퐸 ∶ 퐾]
+푚
+푎푚−1.
+The eigenvalue 휆푗,푠 of 푋◦푃 훾
+푚 has minimal polynomial 휑푗,푚(푡) over Q(훾). Applying Property (v) to 휆푗,푠 ∈
+퐹푗, Equation (16) gives
+Tr퐹푗∕Q(훾)(휆푗,푠) =
+[퐹푗 ∶ Q(훾)]
+푚
+(
+훾 + 휃푗
+)
+.
+(19)
+Consider the smallest extension field 푀 of 퐹푗 that contains 퐹0, … , 퐹푛−1. For 푗 = 0, … , 푛 − 1, 푀∕퐹푗 is a
+Galois extension. It follows from Properties (ii) and (iv) and Equation (19) that
+Tr푀∕Q(훾)(휆푗,푠) = Tr퐹푗∕Q(훾)
+(
+[푀 ∶ 퐹푗]휆푗,푠
+)
+= [푀 ∶ 퐹푗]
+[퐹푗 ∶ Q(훾)]
+푚
+(
+훾 + 휃푗
+)
+= [푀 ∶ Q(훾)]
+푚
+(
+훾 + 휃푗
+)
+.
+(20)
+18
+
+Theorem 5.8. Let 푋 be a Hermitian circulant on 푛 vertices that admits universal perfect state transfer with
+eigenvalues given in Theorem 5.5. If 휃0, … , 휃푛−1 ∈ Q and 훾 is transcendental then, for any positive integer
+푚, the rooted product 푋◦푃 훾
+푚 has multiple pretty good state transfer on the set {(푥0, ℎ), (푥1, ℎ), … , (푥푛−1, ℎ)},
+for 1 ≤ ℎ ≤ 푚.
+Proof. For ℎ = 1, … , 푚, 푋◦푃 훾
+푚 has an automorhism that maps (푥푎, ℎ) to (푥푎+1, ℎ), for 푎 ∈ Z푛. It is sufficient
+to show that there is pretty good state transfer from (푥0, ℎ) to (푥1, ℎ). By Lemma 5.6, (푥0, ℎ) and (푥1, ℎ) are
+strongly cospectral with quarrels
+푞푗,푠
+(
+(푥0, ℎ), (푥1, ℎ)
+)
+= 2휋푗
+푛 ,
+for 푗 = 0, … , 푛 − 1 and 푠 = 1, … , 푚.
+To show the Theorem 2.4 (ii) holds, consider integers 푙푗,푠’s satisfying
+푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠휆푗,푠 = 0.
+(21)
+We apply the trace from 푀 to Q(훾) to both sides. Applying Theorem 5.7 (iii) and Equation (20), Equa-
+tion (21) implies
+푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠(훾 + 휃푗) = 훾
+(푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠
+)
++
+푛−1
+∑
+푗=0
+휃푗
+( 푚
+∑
+푠=1
+푙푗,푠
+)
+= 0.
+Since 훾 is transcendental and ∑
+푗 휃푗
+(∑
+푠 푙푗,푠
+) ∈ Q, Equation (21) is equivalent to
+푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠 = 0
+(22)
+and
+푛−1
+∑
+푗=0
+휃푗
+( 푚
+∑
+푠=1
+푙푗,푠
+)
+= 0.
+(23)
+Recall 휃푗 = 훼 + 훽(푗ℎ + 푐푗푛) where gcd(ℎ, 푛) = 1. Equations (22) and (23) imply
+푛−1
+∑
+푗=0
+(푗ℎ + 푐푗푛)
+( 푚
+∑
+푠=1
+푙푗,푠
+)
+= 0.
+Since gcd(ℎ, 푛) = 1, we have
+푛−1
+∑
+푗=0
+푗
+푚
+∑
+푠=1
+푙푗,푠 = 0
+(mod 푛).
+If Equations (22) and (23) hold then, for any 훿 ∈ R,
+푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠
+(
+푞푗,푠
+(
+(푥0, ℎ), (푥1, ℎ)
+)
++ 훿
+)
+= 2휋
+푛
+(푛−1
+∑
+푗=0
+푗
+푚
+∑
+푠=1
+푙푗,푠
+)
++ 훿
+(푛−1
+∑
+푗=0
+푚
+∑
+푠=1
+푙푗,푠
+)
+= 0
+(mod 2휋).
+By Theorem 2.4, pretty good state transfer occurs from (푥0, ℎ) to (푥1, ℎ), for ℎ = 1, … , 푚.
+19
+
+Remark 5.9.
+• Putting a transcendental weight 훾 on the loops is sufficient for 휑0,푚(푡), … , 휑푛−1,푚(푡) to be irreducible
+over Q(훾). Theorem 5.8 holds for irrational number 훾 as long as 휑0,푚(푡), … , 휑푛−1,푚(푡) are irreducible
+over Q(훾).
+• If we move the loops from the (푥푎, 1) to (푥푎, 푚), for 푎 = 0, … , 푛−1, then a similar argument shows that
+the resulting graph has multiple pretty good state transfer on the set {(푥0, ℎ), (푥1, ℎ), … , (푥푛−1, ℎ)}, for
+ℎ = 1, … , 푚.
+Acknowledgements
+This project was completed under the 2021 Fields Undergraduate Summer Research Program which provided
+support for A. Acuaviva, S. Eldridge, M. How and E. Wright. C. Godsil gratefully acknowledges the support
+of the Natural Sciences and Engineering Council of Canada (NSERC) Grant No. RGPIN-9439. A. Chan is
+grateful for the support of the NSERC Grant No. RGPIN-2021-03609.
+References
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+Towards AI-controlled FES-restoration of arm
+movements: Controlling for progressive muscular
+fatigue with Gaussian state-space models
+Nat Wannawas
+Dept. of Bioengineering, Imperial College London
+London, UK
+nat.wannawas18@imperial.ac.uk
+A. Aldo Faisal
+Dept. of Bioengineering & Dept. of Computing,
+Imperial College London, London, UK
+Chair of Digital Health & Data Science, University of Bayreuth
+Bayreuth, Germany
+aldo.faisal@imperial.ac.uk
+Abstract—Reaching disability limits an individual’s ability in
+performing daily tasks. Surface Functional Electrical Stimulation
+(FES) offers a non-invasive solution to restore lost ability.
+However, inducing desired movements using FES is still an
+open engineering problem. This problem is accentuated by the
+complexities of human arms’ neuromechanics and the variations
+across individuals. Reinforcement Learning (RL) emerges as
+a promising approach to govern customised control rules for
+different settings. Yet, one remaining challenge of controlling FES
+systems for RL is unobservable muscle fatigue that progressively
+changes as an unknown function of the stimulation, thereby
+breaking the Markovian assumption of RL.
+In this work, we present a method to address the unobservable
+muscle fatigue issue, allowing our RL controller to achieve higher
+control performances. Our method is based on a Gaussian State-
+Space Model (GSSM) that utilizes recurrent neural networks
+to learn Markovian state-spaces from partial observations. The
+GSSM is used as a filter that converts the observations into
+the state-space representation for RL to preserve the Markovian
+assumption. Here, we start with presenting the modification of
+the original GSSM to address an overconfident issue. We then
+present the interaction between RL and the modified GSSM,
+followed by the setup for FES control learning. We test our RL-
+GSSM system on a planar reaching setting in simulation using
+a detailed neuromechanical model. The results show that the
+GSSM can help improve the RL’s control performance to the
+comparable level of the ideal case that the fatigue is observable.
+Index Terms—Functional Electrical Stimulation, FES, Gaus-
+sian State-Space Model, Reinforcement Learning, Arm Motions
+I. INTRODUCTION
+Yearly, strokes and spinal cord injuries have left individuals
+around the world with paralysis. Upper body paralysis, one
+of the most commonly found following incidents, causes the
+dysfunction of arm movements and severely affect the individ-
+uals’ abilities in performing daily tasks. Functional Electrical
+Stimulation (FES), a technique that uses electrical signals to
+induce muscle contraction, offers a solution for restoring the
+movements. Yet, controlling FES to induce desired movements
+We acknowledge funding from the Royal Thai Government Scholarship to
+NW and a UKRI Turing AI Fellowship to AAF.
+is challenging. One challenge is that each individual requires
+customised stimulation to induce a certain movement. This
+causes difficulties in designing a control method that works
+across different individuals without intensive, manual config-
+urations. Another challenge is that the muscle’s responses to
+the FES change over time because of muscular fatigue. Since
+the fatigue level can not be directly measured, it is difficult for
+a controller to maintain its performance over extended periods.
+Several methods that can automatically find customised
+stimulation have been investigated. One of those is Reinforce-
+ment Learning (RL), a machine learning algorithm with a
+learning agent (RL agent) that learns to control an environment
+through interaction. The successes of RL in controlling body
+movements have been presented in several scenarios: cycling
+[1], walking [2], arm movements [3]–[7]. Additionally, [7]
+shows that RL can deal with fatigue to a certain degree; yet,
+the performance drop is still inevitable in many cases.
+Different approaches have been employed to deal with
+muscular fatigue. A widely used approach is to record elec-
+tromyogram (EMG) or mechanomyogram (MMG) from which
+muscle force can be estimated [8]–[11]. Although this ap-
+proach could be straightforward, the successes are, currently,
+limited to a few types of movements such as knee extension
+[10], [11] and cycling [8], [9]. Additionally, it requires sensors
+which can be difficult to set up. Approaches that exploit
+the periodic nature of the movements such as walking are
+used in [12], [13]. However, these may not be suitable to be
+used in controlling arbitrary arm movements. Our previous
+work [6] explores an approach that does not use dedicated
+sensors and can be applied to arbitrary movements. The
+approach uses a recurrent neural network (RNN) to encode
+the history of observations and provide additional information
+to the RL agent. This strategy can control arbitrary single-
+joint movements in the real world. However, its capability in
+multiple-joint cases is limited.
+In this work, we present an AI-based system for con-
+trolling FES that can induce arbitrary desired movements
+and can maintain performance under progressive muscular
+fatigue. Our system uses the combination of an RNN-based
+arXiv:2301.04005v1  [eess.SY]  10 Jan 2023
+
+Gaussian state-space model (GSSM) that learns Markovian
+state-representations and RL that learns the control policies
+on the representation spaces. In simple terms, the GSSM
+here functions as a filter that provides insight information
+of the systems’ states to the RL agents, allowing the agents
+to select better actions. Compared to our previous work [6],
+this system is more powerful and capable of leaning the
+complex dynamics of multiple-joint movements. Additionally,
+it produces probabilistic transition functions that can be useful,
+for example, for model-based RL.
+We present the details of our RL-GSSM system and the
+setup for controlling arbitrary movements in the Methods
+section. We also provide the modification of the original
+GSSM [14] to address an overconfident issue. We demonstrate
+our system in a planar arbitrary reaching setting using a
+detailed neuromechanical model and show that our system can
+achieve and maintain the control performance at the same level
+as the ideal case in which muscle fatigue is observable.
+II. METHODS
+A. Gaussian State-Space Models (GSSM)
+Here, GSSM functions as a filter that converts an observable
+environment’s state vector (ot) into a state-representation vec-
+tor (xt) which contains the information of the system’s hidden
+states. Our GSSM is based on [14] whose main components
+are an RNN-based filter (fF ilter) and a transition function
+(fT ran). The filter converts ot into xt through a process
+described as follows. The process starts at the zeroth time
+step (t = 0) with the initialisation of the RNN’s hidden states
+(h0) and state representations (x0). x0 is then concatenated
+with the initial action vector ainit and is passed through Ws,
+a small multilayer perceptron (MLP). This step is mathemati-
+cally expressed as hx,t=0 = Ws([x0; a0]T ). Meanwhile, the
+RNN observes the environment’s states o0 and updates its
+hidden state to ht=1. hx,t=0 and ht=1 are then combined
+as hc,t=1 =
+1
+2 tanh(hx,t=0 + ht=1). Next, hc,t=1 is passed
+through Wx which is an MLP that outputs the distribution of
+xt. The following time steps repeat this process but start with
+the sampled xt and actual actions at. The trajectory of xt,
+denoted as x0:T , is obtained by repeating this process through
+the whole trajectory of observations o0:T . For future notation,
+RNN, Wh, and Wx are referred collectively as fF ilter.
+The GSSM is trained using the trajectory of observations
+(o0:T ) as follows. The training process starts with using fF ilter
+to sample x0:T corresponding to o0:T . Next, we reconstruct the
+observations by passing the sampled x0:T through the obser-
+vation mapping function Wg, expressed as k0:T = Wg(x0:T ).
+The parameters of fF ilter are optimised through gradient
+descent to minimise the following loss functions. The first loss
+function is the likelihood between k0:T and o0:T , expressed
+as llik = �T
+t=1 p(ot|µk,t, Σk,t), where µk,t and Σk,t are
+the mean and covariance of the reconstructed observations,
+respectively. The second loss function is the KL divergence
+between the x0:T distribution sampled by fF ilter and those
+predicted by fT ran, expressed as
+lDKL =
+T
+�
+t=2
+DKL[fF ilter(xt−1, o0:t)||fT ran(xt−1)].
+Intuitively, this loss function encourages the filter-generated
+distribution of xt, pf(xt), to have a Markovian structure,
+i.e, pf(xt|xt−1, o0:t) = p(xt|xt−1). Note that the observation
+history o0:t−1 is encoded in the RNN’s hidden states.
+In the original model [14], fT ran is represented by a neural
+network that directly outputs the means and variances of xt.
+This network produces overconfidence in the learned transition
+function. To mitigate this issue, we replace that network
+with the ensemble of neural networks with randomised prior
+functions (RP-Ensemble) [15]. The predictive means and vari-
+ances are computed by fitting Gaussian distributions to the
+ensemble’s outputs.
+B. Generic RL-GSSM for controlling arbitrary movements
+Reinforcement Learning (RL) learns a task through reward
+signals collected from interactions with an environment. The
+interactions occur in a discrete-time fashion, starting with
+the agent observing the environment’s state st and selecting
+an action at based on its policy π. The action causes the
+environment to be in a new state st+1. The agent then receives
+an immediate reward rt and observes the new state. This
+interaction experience is collected as a tuple (st, at, rt, st+1)
+which is stored in a replay buffer D. This tuple is used to learn
+an optimal policy π∗ that maximises a return R–the sum of
+discounted immediate rewards.
+The introduction of GSSM into the system causes few
+changes in the typical RL learning process. To avoid confusing
+notation, we hereafter use st to denote RL state vectors.
+Fig.1 shows the overview diagram of our RL-GSSM system.
+The system has two phases–interaction and updating phases–
+described as follows. At each time step in the interaction
+phase, fF ilter observes ot, updates the RNN’s hidden states,
+and generates state-representations xt. The agent then selects
+an action at based on st = [ot; xt; ct]T , where ct is a control
+target at time t. The action affects the environment, the system
+moves into the next time step, and the process repeats. The
+interactions are stored as ([ot; ct]T , at, rt, [ot+1); ct+1)]T ) in
+a Trajectory Buffer.
+The updating phase begins with drawing sampled trajec-
+tories (˜o0:T ) from the Trajectory Buffer and using them to
+update the GSSM. After that, the updated fF ilter is used to
+generate new trajectories of st corresponding to ˜o0:T . The
+new st trajectories are then converted into new RL experience
+tuples stored in a typical Replay Buffer, and the RL agent is
+updated following a typical method.
+C. RL-GSSM setup for controlling planar movements
+The environment here is a neuromechanical model built in
+OpenSim. The model has a human arm placed on an arm
+support that moves with low friction on a table Fig.2b. The
+
+Fig. 1. (a) Diagram showing the overview of our RL-GSSM system. The dash blue line splits RL and GSSM. The GSSM’s parts in yellow boxes are excluded
+during the interaction phase. This phase starts with the initialisation (on the left) and evolves as follows. At the time step t, The previous action at−1 are
+appended to the state-representations of the previous time step xt−1. The Filter then combines the appended vector with the incoming observation ot and
+samples the state-representations of the current time step xt. The average of xt, denoted as ¯xt, is concatenated with ot and a control target ct and become an
+RL’s state vector st. The interaction data are stored in Trajectory Buffer. (b) Diagram showing the overview of the training phase that begins with sampling
+the stored trajectories and updating GSSM. The updated Filter is then used to generate new RL’s experience tuples which are used to update the RL agent.
+model has 6 muscles; 4 muscles labelled in the figure are stim-
+ulated. The muscles are fatigued progressively as a function
+of the stimulation (see [1] for more details). The observable
+environment states are the angle and angular velocities of the
+shoulder and elbow (ot = [θs,t; θe,t; ˙θs,t; ˙θe,t]T ).
+The RL algorithm of choice is soft actor-critic [16]. Both
+actor and critic are parameterised by fully-connected neural
+networks with two hidden layers. The actor’s output layer has a
+sigmoid activation function to squash the outputs within [0, 1].
+The RL task here is to apply the muscle stimulation to move
+the arm to the desired poses which are specified by target
+joint angles–shoulder and elbow (θtar,t). The state vector st
+is [ot; xt; θtar,t]T . The action vector at comprises normalised
+stimulation intensities (i ∈ [0, 1]) of the stimulated muscles.
+The immediate reward rt is simply computed using the square
+error and action penalty as rt = −(θt − θtar,t)2 − Σn
+i=0ai
+n
+,
+where n is the number of stimulated muscles.
+The training is episodic. Each episode has 100 time steps
+with a 100 ms time step size. The episodes begin at random
+poses, targets, and fatigue levels. A new random target is
+assigned at the 50th time step. Every 5 training episodes, the
+control performances are evaluated in rmse measure on 50 test
+episodes with the same settings as the training episodes.
+III. RESULTS
+A. Ensemble transition function
+We replace fT rans of the original model [14], denoted as
+fT r,Ori, with RP-Ensemble, denoted as fT r,Ens, to address the
+overconfidence issue. We test both models on a benchmarking
+function–Kink [17]. Fig.2a shows the learned transitions. Both
+fT r,Ori and fT r,Ens produce good predictive means. However,
+fT r,Ori is overconfident as presented by low predictive vari-
+ances at the locations where the data, represented by x marks,
+are absent. In contrast, fT r,Ens has higher predictive variances
+at those locations.
+B. Controlling planar arm movements
+We train our RL-GSSM to control planar arm movements
+under progressive muscular fatigue through muscle stimula-
+tion. We explore 3 cases: the 1) RL-ideal and RL-vanilla cases
+where the fatigue is observable and unobservable, respectively;
+and 3) RL-GSSM case. The RL agents are trained for 100
+episodes in all cases; the training is repeated 10 times.
+Fig.2c shows the performance evaluations in rmse measure
+along the training. RL-vanilla’s performance has the steepest
+improvement at the beginning but stagnates at the worst
+levels. RL-GSSM’s curve, compared to RL-ideal, has higher
+standard deviations in the early period because the agents have
+to simultaneously learn the controls and follow the not-yet-
+converged GSSM. RL-GSSM’s performance improves slightly
+slower but can reach the same level in 100 episodes.
+Fig.3 shows the control behaviours in tracking an arbitrary
+trajectory. The agents can produce good tracking in all cases.
+The grey circles highlight good comparison points. Both RL-
+ideal (Fig.3a) and RL-GSSM (Fig.3c) can bring the shoulder
+and elbows to the [45◦, 45◦] targets anytime when requested.
+RL-vanilla, however, tends to lose its performance in the
+second half as the actual angles increasingly deviate from the
+targets (Fig.3b). Fig.3d-f show the stimulation (solid lines) and
+%maximum force that the muscles can produce (dash lines).
+The %maximum force decreases over time as the stimulation
+induces muscular fatigue. Compared to RL-ideal (Fig.3d), RL-
+vanilla (Fig.3e) over stimulates and causes the rapid declines
+of the muscle forces. The declines in RL-GSSM and RL-ideal
+cases are at the same rate in average. RL-GSSM’s stimulation
+has small noises along the session.
+
+GSSM
+RL
+Training
+Xo
+UpdatableO
+Network
+Q
++
+RP
+(frozen)
+ainit
+Filter
+Initialisation
+RNN
+(ho)Fig. 2. (a) The learnt kink function of the (left) original GSSM and (right) the GSSM with RP-Ensemble transition function. (b) Neuromechanical model of
+planar arm movement built in OpenSim. (c) The control performances evaluated along the training. The shades show the standard deviations of 10 runs.
+Fig. 3. Control behaviours in tracking an arbitrary target trajectory. (a-c) The plots showing the targets (dash) and the actual angles (solid) are achieved in (a)
+RL − ideal, (b) RL − vanilla, and (c) RL − GSSM cases. (d-f) %maximum stimulation that the RL agents apply on the muscles (solid) and %maximum
+forces that the muscles can produce (dash). The %maximum forces decrease in response to the muscular fatigue induced by the stimulation.
+IV. CONCLUSIONS
+We present a AI-based approach for controlling FES under
+progressive muscular fatigue. Our RL-GSSM approach uses
+RL to learn the control policies and GSSM, modified to
+address the overconfidence issue, to provide Makovian state-
+representations to the RL. We demonstrate our approach to
+controlling arbitrary planar arm movements using a detailed
+neuromechanical model. We show that our RL-GSSM can
+achieve and maintain its control performances at the same level
+as the ideal case where the fatigue is observable.
+REFERENCES
+[1] N. Wannawas, M. Subramanian, and A. A. Faisal, “Neuromechanics-
+based deep reinforcement learning of neurostimulation control in fes
+cycling,” in Intl. IEEE/EMBS Conf. on Neural Engineering (NER), 2021.
+[2] A. Anand et al., “A deep reinforcement learning based approach towards
+generating human walking behabior with a neuromuscular model,” in
+19th Intl. Conf. on Humanoid Robots, 2019.
+[3] P. Thomas et al., “Creating a reinforcement learning controller for
+functional electrical stimulation of a human arm,” in 14th Yale Workshop
+on Adaptive and Learning Systems, 2008.
+[4] K. M. Jagodnik et al., “Human-like rewards to train a reinforcement
+learning controller for planar arm movement,” IEEE Trans on Human-
+Machine Systems, vol. 46, pp. 723–733, 10 2016.
+[5] D. N. Wolf, Z. A. Hall, and E. M. Schearer, “Model learning for control
+of a paralyzed human arm with functional electrical stimulation,” in
+IEEE Intl. Conf. on Robotics and Automation (ICRA), 2020, p. 10148.
+[6] N. Wannawas, A. Shafti, and A. A. Faisal, “Neuromuscular reinforce-
+ment learning to actuate human limbs through fes,” in IFESS22, 2022.
+[7] J. Abreu et al., “Deep reinforcement learning for control of time-varying
+musculoskeletal systems with high fatigability: a feasibility study,” in
+IEEE Trans. Neural Sys. and Rehab. Eng., 2022.
+[8] B. Woods, M. Subramanian, A. Shafti, and A. A. Faisal, “Mechanomyo-
+graphy based closed-loop functional electrical stimulation cycling sys-
+tem,” in 7th IEEE Intl. Conf. on Biomed. Robotics and Biomechatronics,
+vol. 2018-Augus.
+IEEE, 8 2018, pp. 179–184.
+[9] M. Islam et al., “Mechanomyography responses characterize altered
+muscle function during electrical stimulation-evoked cycling in individ-
+uals with spinal cord injury,” Clinical Biomechanics, vol. 58, 2018.
+[10] J. Naeem et al., “Electrical stimulator with mechanomyography-based
+real-time monitoring, muscle fatigue detection, and safety shut-off: A
+pilot study,” Biomedizinische Technik, vol. 65, 2020.
+[11] E. Krueger et al., “Neuromuscular fatigue detection by mechanomyogra-
+phy in people with complete spinal cord injury,” Research on Biomedical
+Engineering, vol. 36, pp. 203–212, 2020.
+[12] A. J. Del-Ama, ´Angel Gil-Agudo, J. L. Pons, and J. C. Moreno,
+“Hybrid fes-robot cooperative control of ambulatory gait rehabilitation
+exoskeleton,” J. NeuroEngineering and Rehabilitation, vol. 11, 2014.
+[13] K. H. Ha et al., “An approach for the cooperative control of fes with a
+powered exoskeleton during level walking for persons with paraplegia,”
+IEEE Trans on Neural Sys. and Rehab. Eng., vol. 24, 2016.
+[14] R. G. Krishnan, U. Shalit, and D. Sontag, “Structured inference networks
+for nonlinear state space models,” in AAAI, 2017.
+[15] I. Osband, J. Aslanides, and A. Cassirer, “Randomized prior functions
+for deep reinforcement learning,” in NIPS, 2018.
+[16] T. Haarnoja et al., “Soft actor-critic algorithms and applications,”
+arXiv:1812.05905v2 [cs.LG], 2019.
+[17] A. D. Ialongo et al., “Overcoming mean-field approximations in recur-
+rent gaussian process models,” in 36th ICML, 2019.
+
+30 -
+Original
+Ensemble
+Obs-Fatigue
+Not-Obs-Fatigue
+25 
+GSSM
+Deltoid
+-1
+Posterior
+Pectoralis major C
+E 20 +
+×-2
+Brachialis
+-3
+Table
+-4
+Triceps
+10
+True function
+True function
+Medial
+5
+Arm
+Learned function
+Learned function
+Support
+-6
+4
+2
+0
+4
+2
+0
+5
+6
+-6
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Xt 1
+Xt-1
+a
+Training Episode
+cRL-ideal (observablefatigue)
+RMSE: 7.02 °
+RL-vanilla (unobservable fatigue) RMSE: 8.05
+RL-GSSM
+RMSE: 6.84
+100
+b
+c
+80
+60
+Angle [
+40
+20
+ Shoulder
+Elbow
+ Shoulder
+Elbow
+Shoulder
+Elbow
+0
+Biceps
+Triceps
+Pect. Maj.
+Deltoid Post.
+= Biceps
+Triceps
+Pect. Maj.
+Deltoid Post.
+Biceps
+Triceps
+Pect. Maj.
+Deltoid Post.
+Force
+d
+e
+Stimulation (%)
+80
+Max. Muscle F
+60
+40 
+20
+&
+0
+0
+10
+15 
+20
+25
+30
+35
+40
+45
+50
+55
+60 0
+5
+10
+15
+20
+25
+30
+35
+40
+45
+50
+55
+60 0
+5
+10
+15
+25
+30
+35
+40
+45
+55
+60
+time [s]
+time [s]
+time [s]

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+page_content='Towards AI-controlled FES-restoration of arm  movements: Controlling for progressive muscular  fatigue with Gaussian state-space models  Nat Wannawas  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
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+page_content=' Aldo Faisal  Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' of Bioengineering & Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' of Computing,  Imperial College London, London, UK  Chair of Digital Health & Data Science, University of Bayreuth  Bayreuth, Germany  aldo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='faisal@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
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+page_content='uk  Abstract—Reaching disability limits an individual’s ability in  performing daily tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Surface Functional Electrical Stimulation  (FES) offers a non-invasive solution to restore lost ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  However, inducing desired movements using FES is still an  open engineering problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This problem is accentuated by the  complexities of human arms’ neuromechanics and the variations  across individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Reinforcement Learning (RL) emerges as  a promising approach to govern customised control rules for  different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Yet, one remaining challenge of controlling FES  systems for RL is unobservable muscle fatigue that progressively  changes as an unknown function of the stimulation, thereby  breaking the Markovian assumption of RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  In this work, we present a method to address the unobservable  muscle fatigue issue, allowing our RL controller to achieve higher  control performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Our method is based on a Gaussian State-  Space Model (GSSM) that utilizes recurrent neural networks  to learn Markovian state-spaces from partial observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  GSSM is used as a filter that converts the observations into  the state-space representation for RL to preserve the Markovian  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Here, we start with presenting the modification of  the original GSSM to address an overconfident issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We then  present the interaction between RL and the modified GSSM,  followed by the setup for FES control learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We test our RL-  GSSM system on a planar reaching setting in simulation using  a detailed neuromechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The results show that the  GSSM can help improve the RL’s control performance to the  comparable level of the ideal case that the fatigue is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Index Terms—Functional Electrical Stimulation, FES, Gaus-  sian State-Space Model, Reinforcement Learning, Arm Motions  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' INTRODUCTION  Yearly, strokes and spinal cord injuries have left individuals  around the world with paralysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Upper body paralysis, one  of the most commonly found following incidents, causes the  dysfunction of arm movements and severely affect the individ-  uals’ abilities in performing daily tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Functional Electrical  Stimulation (FES), a technique that uses electrical signals to  induce muscle contraction, offers a solution for restoring the  movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Yet, controlling FES to induce desired movements  We acknowledge funding from the Royal Thai Government Scholarship to  NW and a UKRI Turing AI Fellowship to AAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' One challenge is that each individual requires  customised stimulation to induce a certain movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This  causes difficulties in designing a control method that works  across different individuals without intensive, manual config-  urations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Another challenge is that the muscle’s responses to  the FES change over time because of muscular fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Since  the fatigue level can not be directly measured, it is difficult for  a controller to maintain its performance over extended periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Several methods that can automatically find customised  stimulation have been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' One of those is Reinforce-  ment Learning (RL), a machine learning algorithm with a  learning agent (RL agent) that learns to control an environment  through interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The successes of RL in controlling body  movements have been presented in several scenarios: cycling  [1], walking [2], arm movements [3]–[7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Additionally, [7]  shows that RL can deal with fatigue to a certain degree;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' yet,  the performance drop is still inevitable in many cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Different approaches have been employed to deal with  muscular fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' A widely used approach is to record elec-  tromyogram (EMG) or mechanomyogram (MMG) from which  muscle force can be estimated [8]–[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Although this ap-  proach could be straightforward, the successes are, currently,  limited to a few types of movements such as knee extension  [10], [11] and cycling [8], [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Additionally, it requires sensors  which can be difficult to set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Approaches that exploit  the periodic nature of the movements such as walking are  used in [12], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' However, these may not be suitable to be  used in controlling arbitrary arm movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Our previous  work [6] explores an approach that does not use dedicated  sensors and can be applied to arbitrary movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  approach uses a recurrent neural network (RNN) to encode  the history of observations and provide additional information  to the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This strategy can control arbitrary single-  joint movements in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' However, its capability in  multiple-joint cases is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  In this work, we present an AI-based system for con-  trolling FES that can induce arbitrary desired movements  and can maintain performance under progressive muscular  fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Our system uses the combination of an RNN-based  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='04005v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='SY]  10 Jan 2023  Gaussian state-space model (GSSM) that learns Markovian  state-representations and RL that learns the control policies  on the representation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' In simple terms, the GSSM  here functions as a filter that provides insight information  of the systems’ states to the RL agents, allowing the agents  to select better actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Compared to our previous work [6],  this system is more powerful and capable of leaning the  complex dynamics of multiple-joint movements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Additionally,  it produces probabilistic transition functions that can be useful,  for example, for model-based RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  We present the details of our RL-GSSM system and the  setup for controlling arbitrary movements in the Methods  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We also provide the modification of the original  GSSM [14] to address an overconfident issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We demonstrate  our system in a planar arbitrary reaching setting using a  detailed neuromechanical model and show that our system can  achieve and maintain the control performance at the same level  as the ideal case in which muscle fatigue is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' METHODS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Gaussian State-Space Models (GSSM)  Here, GSSM functions as a filter that converts an observable  environment’s state vector (ot) into a state-representation vec-  tor (xt) which contains the information of the system’s hidden  states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Our GSSM is based on [14] whose main components  are an RNN-based filter (fF ilter) and a transition function  (fT ran).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The filter converts ot into xt through a process  described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The process starts at the zeroth time  step (t = 0) with the initialisation of the RNN’s hidden states  (h0) and state representations (x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' x0 is then concatenated  with the initial action vector ainit and is passed through Ws,  a small multilayer perceptron (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This step is mathemati-  cally expressed as hx,t=0 = Ws([x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' a0]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Meanwhile, the  RNN observes the environment’s states o0 and updates its  hidden state to ht=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' hx,t=0 and ht=1 are then combined  as hc,t=1 =  1  2 tanh(hx,t=0 + ht=1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Next, hc,t=1 is passed  through Wx which is an MLP that outputs the distribution of  xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The following time steps repeat this process but start with  the sampled xt and actual actions at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The trajectory of xt,  denoted as x0:T , is obtained by repeating this process through  the whole trajectory of observations o0:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' For future notation,  RNN, Wh, and Wx are referred collectively as fF ilter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The GSSM is trained using the trajectory of observations  (o0:T ) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The training process starts with using fF ilter  to sample x0:T corresponding to o0:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Next, we reconstruct the  observations by passing the sampled x0:T through the obser-  vation mapping function Wg, expressed as k0:T = Wg(x0:T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The parameters of fF ilter are optimised through gradient  descent to minimise the following loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The first loss  function is the likelihood between k0:T and o0:T , expressed  as llik = �T  t=1 p(ot|µk,t, Σk,t), where µk,t and Σk,t are  the mean and covariance of the reconstructed observations,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The second loss function is the KL divergence  between the x0:T distribution sampled by fF ilter and those  predicted by fT ran, expressed as  lDKL =  T  �  t=2  DKL[fF ilter(xt−1, o0:t)||fT ran(xt−1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Intuitively, this loss function encourages the filter-generated  distribution of xt, pf(xt), to have a Markovian structure,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='e, pf(xt|xt−1, o0:t) = p(xt|xt−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Note that the observation  history o0:t−1 is encoded in the RNN’s hidden states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  In the original model [14], fT ran is represented by a neural  network that directly outputs the means and variances of xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  This network produces overconfidence in the learned transition  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' To mitigate this issue, we replace that network  with the ensemble of neural networks with randomised prior  functions (RP-Ensemble) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The predictive means and vari-  ances are computed by fitting Gaussian distributions to the  ensemble’s outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Generic RL-GSSM for controlling arbitrary movements  Reinforcement Learning (RL) learns a task through reward  signals collected from interactions with an environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  interactions occur in a discrete-time fashion, starting with  the agent observing the environment’s state st and selecting  an action at based on its policy π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The action causes the  environment to be in a new state st+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The agent then receives  an immediate reward rt and observes the new state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This  interaction experience is collected as a tuple (st, at, rt, st+1)  which is stored in a replay buffer D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This tuple is used to learn  an optimal policy π∗ that maximises a return R–the sum of  discounted immediate rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The introduction of GSSM into the system causes few  changes in the typical RL learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' To avoid confusing  notation, we hereafter use st to denote RL state vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='1 shows the overview diagram of our RL-GSSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The system has two phases–interaction and updating phases–  described as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' At each time step in the interaction  phase, fF ilter observes ot, updates the RNN’s hidden states,  and generates state-representations xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The agent then selects  an action at based on st = [ot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' ct]T , where ct is a control  target at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The action affects the environment, the system  moves into the next time step, and the process repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  interactions are stored as ([ot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' ct]T , at, rt, [ot+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' ct+1)]T ) in  a Trajectory Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The updating phase begins with drawing sampled trajec-  tories (˜o0:T ) from the Trajectory Buffer and using them to  update the GSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' After that, the updated fF ilter is used to  generate new trajectories of st corresponding to ˜o0:T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  new st trajectories are then converted into new RL experience  tuples stored in a typical Replay Buffer, and the RL agent is  updated following a typical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RL-GSSM setup for controlling planar movements  The environment here is a neuromechanical model built in  OpenSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The model has a human arm placed on an arm  support that moves with low friction on a table Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (a) Diagram showing the overview of our RL-GSSM system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The dash blue line splits RL and GSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The GSSM’s parts in yellow boxes are excluded  during the interaction phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' This phase starts with the initialisation (on the left) and evolves as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' At the time step t, The previous action at−1 are  appended to the state-representations of the previous time step xt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The Filter then combines the appended vector with the incoming observation ot and  samples the state-representations of the current time step xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The average of xt, denoted as ¯xt, is concatenated with ot and a control target ct and become an  RL’s state vector st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The interaction data are stored in Trajectory Buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (b) Diagram showing the overview of the training phase that begins with sampling  the stored trajectories and updating GSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The updated Filter is then used to generate new RL’s experience tuples which are used to update the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  model has 6 muscles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' 4 muscles labelled in the figure are stim-  ulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The muscles are fatigued progressively as a function  of the stimulation (see [1] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The observable  environment states are the angle and angular velocities of the  shoulder and elbow (ot = [θs,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' θe,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' ˙θs,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' ˙θe,t]T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The RL algorithm of choice is soft actor-critic [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Both  actor and critic are parameterised by fully-connected neural  networks with two hidden layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The actor’s output layer has a  sigmoid activation function to squash the outputs within [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The RL task here is to apply the muscle stimulation to move  the arm to the desired poses which are specified by target  joint angles–shoulder and elbow (θtar,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The state vector st  is [ot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' xt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' θtar,t]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The action vector at comprises normalised  stimulation intensities (i ∈ [0, 1]) of the stimulated muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The immediate reward rt is simply computed using the square  error and action penalty as rt = −(θt − θtar,t)2 − Σn  i=0ai  n  ,  where n is the number of stimulated muscles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The training is episodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Each episode has 100 time steps  with a 100 ms time step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The episodes begin at random  poses, targets, and fatigue levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' A new random target is  assigned at the 50th time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Every 5 training episodes, the  control performances are evaluated in rmse measure on 50 test  episodes with the same settings as the training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Ensemble transition function  We replace fT rans of the original model [14], denoted as  fT r,Ori, with RP-Ensemble, denoted as fT r,Ens, to address the  overconfidence issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We test both models on a benchmarking  function–Kink [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='2a shows the learned transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Both  fT r,Ori and fT r,Ens produce good predictive means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' However,  fT r,Ori is overconfident as presented by low predictive vari-  ances at the locations where the data, represented by x marks,  are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' In contrast, fT r,Ens has higher predictive variances  at those locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Controlling planar arm movements  We train our RL-GSSM to control planar arm movements  under progressive muscular fatigue through muscle stimula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We explore 3 cases: the 1) RL-ideal and RL-vanilla cases  where the fatigue is observable and unobservable, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  and 3) RL-GSSM case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The RL agents are trained for 100  episodes in all cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' the training is repeated 10 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='2c shows the performance evaluations in rmse measure  along the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RL-vanilla’s performance has the steepest  improvement at the beginning but stagnates at the worst  levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RL-GSSM’s curve, compared to RL-ideal, has higher  standard deviations in the early period because the agents have  to simultaneously learn the controls and follow the not-yet-  converged GSSM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RL-GSSM’s performance improves slightly  slower but can reach the same level in 100 episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3 shows the control behaviours in tracking an arbitrary  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The agents can produce good tracking in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The grey circles highlight good comparison points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Both RL-  ideal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3a) and RL-GSSM (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3c) can bring the shoulder  and elbows to the [45◦, 45◦] targets anytime when requested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  RL-vanilla, however, tends to lose its performance in the  second half as the actual angles increasingly deviate from the  targets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3d-f show the stimulation (solid lines) and  %maximum force that the muscles can produce (dash lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  The %maximum force decreases over time as the stimulation  induces muscular fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Compared to RL-ideal (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3d), RL-  vanilla (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='3e) over stimulates and causes the rapid declines  of the muscle forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The declines in RL-GSSM and RL-ideal  cases are at the same rate in average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' RL-GSSM’s stimulation  has small noises along the session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  GSSM  RL  Training  Xo  UpdatableO  Network  Q  +  RP  (frozen)  ainit  Filter  Initialisation  RNN  (ho)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (a) The learnt kink function of the (left) original GSSM and (right) the GSSM with RP-Ensemble transition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (b) Neuromechanical model of  planar arm movement built in OpenSim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (c) The control performances evaluated along the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The shades show the standard deviations of 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Control behaviours in tracking an arbitrary target trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (a-c) The plots showing the targets (dash) and the actual angles (solid) are achieved in (a)  RL − ideal, (b) RL − vanilla, and (c) RL − GSSM cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' (d-f) %maximum stimulation that the RL agents apply on the muscles (solid) and %maximum  forces that the muscles can produce (dash).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' The %maximum forces decrease in response to the muscular fatigue induced by the stimulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' CONCLUSIONS  We present a AI-based approach for controlling FES under  progressive muscular fatigue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Our RL-GSSM approach uses  RL to learn the control policies and GSSM, modified to  address the overconfidence issue, to provide Makovian state-  representations to the RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We demonstrate our approach to  controlling arbitrary planar arm movements using a detailed  neuromechanical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' We show that our RL-GSSM can  achieve and maintain its control performances at the same level  as the ideal case where the fatigue is observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  REFERENCES  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
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+page_content=' Subramanian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
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+page_content='02 °  RL-vanilla (unobservable fatigue) RMSE: 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='05  RL-GSSM  RMSE: 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='84  100  b  c  80  60  Angle [  40  20  Shoulder  Elbow  Shoulder  Elbow  Shoulder  Elbow  0  Biceps  Triceps  Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Deltoid Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  = Biceps  Triceps  Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Deltoid Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Biceps  Triceps  Pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Maj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Deltoid Post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content='  Force  d  e  Stimulation (%)  80  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}
+page_content=' Muscle F  60  40  20  &  0  0  10  15  20  25  30  35  40  45  50  55  60 0  5  10  15  20  25  30  35  40  45  50  55  60 0  5  10  15  25  30  35  40  45  55  60  time [s]  time [s]  time [s]' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtE2T4oBgHgl3EQfnAjp/content/2301.04005v1.pdf'}

BNE4T4oBgHgl3EQfFAx2/content/tmp_files/2301.04882v1.pdf.txt

@@ -0,0 +1,2112 @@
+ZScribbleSeg: Zen and the Art of Scribble
+Supervised Medical Image Segmentation
+Ke Zhang and Xiahai Zhuang ⋆
+School of Data Science, Fudan University, Shanghai
+zxh@fudan.edu.cn
+Abstract. Curating a large scale fully-annotated dataset can be both
+labour-intensive and expertise-demanding, especially for medical images.
+To alleviate this problem, we propose to utilize solely scribble anno-
+tations for weakly supervised segmentation. Existing solutions mainly
+leverage selective losses computed solely on annotated areas and gen-
+erate pseudo gold standard segmentation by propagating labels to ad-
+jacent areas. However, these methods could suffer from the inaccurate
+and sometimes unrealistic pseudo segmentation due to the insufficient
+supervision and incomplete shape features. Different from previous ef-
+forts, we first investigate the principle of ”good scribble annotations”,
+which leads to efficient scribble forms via supervision maximization and
+randomness simulation. Furthermore, we introduce regularization terms
+to encode the spatial relationship and shape prior, where a new formu-
+lation is developed to estimate the mixture ratios of label classes. These
+ratios are critical in identifying the unlabeled pixels for each class and
+correcting erroneous predictions, thus the accurate estimation lays the
+foundation for the incorporation of spatial prior. Finally, we integrate
+the efficient scribble supervision with the prior into a unified framework,
+denoted as ZScribbleSeg, and apply the method to multiple scenarios.
+Leveraging only scribble annotations, ZScribbleSeg set new state-of-the-
+arts on four segmentation tasks using ACDC, MSCMRseg, MyoPS and
+PPSS datasets.
+Keywords: Medical Image Segmentation· Scribble Supervision· Mix-
+ture Model· Medical Image Analysis
+In recent years, deep neural networks has demonstrated its potential on various
+visual tasks [25]. However, the success of these methods relies on massive anno-
+tations, which require assiduous manual efforts. For medical imaging, the dense
+manual labeling can take several hours to annotate just one image for experi-
+enced doctors, which is both expensive and expertise-demanding [60]. Humor-
+ous efforts have contributed to the area of training segmentation networks with
+weaker annotations [39], including scribbles [27], bounding boxes [34], points [2],
+and image-level labels [35]. Numerous studies have been reported utilizing only
+⋆ Xiahai Zhuang is corresponding author. This work was funded by the National Nat-
+ural Science Foundation of China (Grant No. 61971142 and 62111530195).
+arXiv:2301.04882v1  [cs.CV]  12 Jan 2023
+
+2
+K Zhang & X Zhuang
+image-level labels [15,46,50,45]. These methods mainly rely on large-scale train-
+ing datasets, and tend to underperform on small medical image datasets. On the
+contrary, scribbles are suitable for labeling nested structures and easy to obtain
+in practice. Several works have demonstrated their potential on both semantic
+and medical image segmentation [17,21,27]. Therefore, we propose to investigate
+this specific form of weakly supervised segmentation, which only uses scribble
+annotations for model training.
+Conventionally, scribble annotations are mainly focused on delineating the
+structure of interests [42]. This can be effective in segmenting regular structures,
+i.e., the targets with fixed shape patterns. Hence, this task is also referred to
+as regular structure segmentation. However, such methods could be challenged
+when they were applied to portray the irregular targets with heterogeneous dis-
+tributions, such as pathologies. This is also referred to as irregular (object) seg-
+mentation, which is particularly challenging for the medical tasks with small
+training datasets. Existing scribble learning approaches mainly aim to recon-
+struct complete labels from scribbles, and use the generated pseudo labels for
+model training. These works include 1) label expansion strategies that assume
+the pixels with similar features are likely to be in the same category [16,27],
+and 2) ensemble methods that generate labels by fusing several independent
+predictions [29]. These methods could be susceptible to the label noises intro-
+duced by imprecise segmentation proposals. To overcome this issue, Obukhov
+et al. proposed a regularization loss [32], which exploited the similarity between
+labeled and unlabeled area. Adversarial learning approach has also been applied
+to scribble supervised segmentation [42], by leveraging shape prior provided by
+additional full annotations.
+Scribble supervised segmentation generally suffers from inadequate supervi-
+sion and imbalanced label classes. This leads to poor results, typically of under
+segmentation of target structures, meaning the volumes of segmented structures
+tend to be shrunk, as we shall describe in Section 2.3. To address the problem
+of inadequate supervision, we first investigate the principles of generating ”good
+scribbles”, as a guidance for designing methodologies to augment supervision, as
+well as for generating manual annotations. The aim is to model efficient scrib-
+bles by maximizing the supervision without increasing annotation efforts. Our
+studies demonstrate that the model training benefit from the randomness of
+wide range distributed scribbles and larger proportion of annotated areas. In-
+spired by this, we propose to simulate such types of scribble-annotated images
+as a means of supervision augmentation. This can be achieved via mixup and
+occlusion operations on existing training images, and the supervision augmen-
+tation is coupled with regularization terms penalizing any inconsistency in the
+segmentation results.
+Despite the lack of supervision, the scribble annotations typically have imbal-
+anced annotated label proportions thus biased shape information. This means
+the model cannot accurately capture the global shape of target structures. We
+therefore further propose to correct the problematic prediction using prior-based
+regularization, particularly from the spatial prior. This requires the preceding
+
+ZScribbleSeg
+3
+yet critical step of estimating the mixture proportion (ratio) of each label class
+(referred to as π prior). We hence propose a new algorithm to compute this π
+prior, based on which we develop a spatial loss on the basis of marginal proba-
+bility of pixels belonging to certain label classes and spatial energy. This spatial
+loss is a regularization term aimed to correct the shape of segmentation results.
+The supervision augmentation and prior-based regularization work in a comple-
+mentary way, and both contribute to the stable and robust training on a variety
+of segmentation tasks.
+The proposed scribble supervision-based segmentation method, referred to
+as ZScribbleSeg, extends and generalizes the algorithms in our two preliminary
+works [52,53], and has more scientific significance in the following aspects: Firstly,
+we investigate principles of efficient scribble forms to guide the supervision aug-
+mentation, which have never be reported to the best of our knowledge. Secondly,
+we leverage spatial prior to adjust the predicted probability with computed spa-
+tial energy. Thirdly, we implement a series of extensive experiments on various
+scenarios, including irregular structure segmentation of medical pathology and
+visual object segmentation. The contributions of this paper are summarized as
+follows.
+– We propose a unified framework for scribble-supervised segmentation by
+modeling efficient scribbles, and correcting the network prediction with prior
+regularization, which significantly alleviates the problems of inadequate su-
+pervision and imbalanced label classes.
+– To the best of our knowledge, this is the first work investigating the principles
+of scribble forms. Motivated by the conclusion that network benefits from
+larger and randomly distributed annotation, we model efficient scribbles by
+maximizing supervision and simulating randomness.
+– We propose a novel mechanism to correct the shape of model prediction
+based on prior regularization, including π prior, spatial prior, and shape
+prior. A new algorithm is introduced to estimate π prior, based on which we
+further encode spatial relationship with spatial prior loss.
+– Our approach achieved state-of-the-art performance for weakly-supervised
+segmentation on regular structures from cardiac anatomical imaging, regular
+structures from pathology enhanced imaging, irregular objects of medical
+pathology, and human pose from natural scene.
+The rest of this paper is organized as follows: Section 2 briefly introduces the
+relevant researches. In section 3, we describe the modeling of efficient scribbles
+and computation of prior. Section 4 presents the results of efficiency, ablation,
+and validation study. Finally, we conclude this work in Section 5.
+1
+Related work
+This section provides a brief review of weakly supervised segmentation meth-
+ods. Besides, we describe data augmentation strategies and regularization loss
+functions that closely related to our work.
+
+4
+K Zhang & X Zhuang
+Fig. 1. Roadmap of the proposed ZScribbleSeg framework.
+1.1
+Weakly supervised segmentation
+Recently, a variety of weakly supervised segmentation strategies have been de-
+veloped to reduce the manual annotation efforts [27,2,34,35]. Among them, the
+scribbles are of particular interest for the application to medical image annota-
+tion, given by its advantage in annotating nest structures compared to bounding
+boxes. Current weakly supervised learning methods with image-level annotations
+mainly generate label seeds with Class Activation Map (CAM) [56] at first, and
+then train the network with refined pseudo labels. However, the training of CAM
+requires a large scale of training data labeled with rich visual classes, which is
+not practical in clinical applications. Therefore, we propose to investigate the
+scribble supervised segmentation, due to its efficiency and effectiveness in both
+medical and visual scenarios.
+Scribble is a form of sparse annotation that provides labels for a small sub-
+set of pixels in an image [39]. Previous approaches mainly calculate losses for
+annotated pixels. One group of works is designed to expand the annotations
+and reconstruct the full label for network training. However, the expansion of
+labels needs to be achieved through iterative computation, which is particularly
+time-consuming. To alleviate it, several works removed the relabeling process
+and instead adopted conditional random fields to perform the refinement of seg-
+mentation results [9,7,55,40]. However, the common issue is the unstable model
+training caused by noisy pseudo labels.
+
+Principles
+Efficient scribbles
+(Sec 3.2.1)
+(Sec 3.2.2)
+Maximal
+Mixup
+Randomness
+Occlusion
+supervision
+zScribbleSeg
+Lglobal
+ZScribbleNet
+Priors estimation
+(Sec 3.3)
+(Sec 3.4)
+Prior T
+Spatial priors
+Shape priors
+Lshape
+个
+Energy
+RankingZScribbleSeg
+5
+To obtain high-quality pseudo labels and update it throughout the training
+process, Luo et al. [29] proposed to mix the predictions from dual-branch net-
+work as auxiliary pseudo label. This approach has achieved promising results on
+cardiac segmentation, but still susceptible to inaccurate supervisions, especially
+on more challenging tasks with irregular objects. Obukhov et al. [31] introduced
+the Gated CRF loss for unlabeled pixels, which regularizes model training by
+exploiting the structural similarity between labeled and unlabeled data. Other
+works [42,54] included a new module to evaluate the quality of segmentation
+masks, which encourages the predictions to be realistic, but requiring extra full
+annotations.
+1.2
+Data augmentation
+Augmentation methods are investigated to improve the model generalization
+ability, by synthesizing virtual training examples in the vicinity of the training
+dataset [6]. Common strategies include random cropping, rotation, flipping and
+adding noise [5]. Recently, a line of research works have been proposed on Mixup
+augmentation [51,10,49,18,19], which blends two image-label pairs to generate
+new samples for classification tasks. Input Mixup [51] was introduced to perform
+linear interpolation between two images and their labels. Manifold Mixup [43]
+applied the Mixup operation to feature space. Cutout [10] randomly occluded
+a square region of image, and CutMix [49] transplanted the occluded area to
+another image. Kim et al. [18] proposed Puzzle Mix to leverage the saliency
+and local statistics to facilitate image combination. Comixup [19] extended this
+concept from two images to multiple images.
+For medical image analysis, Mixup methods have been adopted for image
+segmentation [8] and object detection tasks [44]. Although mixup operation may
+generate unrealistic samples, mixed soft labels can provide rich information and
+improve the model performance on semi-supervised segmentation [8].
+1.3
+Regularization losses
+Neural networks are used to perform pixel-wise image segmentation, typically
+trained with cross entropy or Dice loss, which computes loss for each pixel in-
+dependently. To predict segmentation coherent in the global sense [22], several
+methods are proposed to regularize the model training. Here, we focus on the
+consistency regularization and π prior regularization that most relevant to our
+work.
+The consistency regularization leverages the fact that the perturbed versions
+of the same image patch should have the consistent segmentation. A series of
+researches have been conducted on consistency regularization [57,23,41,33]. For
+semi-supervised learning, regularization is applied to the augmented versions of
+the input image by requiring consistency to obtain stable predictions for unla-
+beled images [23,41,33].
+
+6
+K Zhang & X Zhuang
+Fig. 2. Overview of the training losses for the proposed ZScribbleNet, which consists
+of modeling of efficient scribbles and computation of priors. The scribble modeling
+includes mixup augmentation, regularized with global consistency (Lglobal). The priors
+have three, i.e., class mixture ratios (π), spatial prior and shape prior, which contribute
+to spatial prior loss (Lspatial) and shape prior loss (Lshape). Note that spatial prior loss
+is complementary with the partial cross entropy loss (Lpce) which is solely calculated
+for labeled pixels.
+The proposed regularization of π prior is inspired from the binary mixture
+proportion estimation [3,14,37], which was originally designed for binary (two-
+class) positive unlabeled learning [11,12,20]. For multi-class segmentation, the
+mixture ratios of classes are both imbalanced and inter-dependent, which cannot
+be solved by existing binary estimation methods.
+2
+Method
+2.1
+Overview
+Problem Setup: This work investigates the scenario of scribble supervised seg-
+mentation, where the training images are solely annotated with a small number
+of pixels, via scribbles, for each label class.
+Strategy: Instead of solely focusing on techniques of weak supervision, we first
+investigate different forms of scribbles to derive principles of efficient scribbles,
+i.e., maximal supervision without increasing scribble efforts. These principles
+enable effective and robust model training with minimal annotation cost. Then,
+we focus on tackling the major problem of under segmentation, to correct model
+prediction with prior.
+Solution: We develop ZScribbleSeg consisting of (1) modeling efficient scrib-
+bles via supervision maximization and randomness simulation. (2) modeling
+
+Priors
+spatial (Eq.29)
+shape
+L
+(Eq.30)
+π (3.3.2)
+Spatial
+Corrected
+Image 1
+Seg 1
+Scribble 1
+energy 1
+shape 1
+Ranking
+A
+Seg of
+Mixed
+Mixed
+Mixed
+mixed
+Network
+Seg
+scribble
+Image
+image
+π (3.3.2)
+Spatial
+Corrected
+Image 2
+Seg 2
+Scribble 2
+energy 2
+shape 2
+Lshape
+Ranking
+(Eq.30)
+4-
+Priors
+Spatial energy Correction Scribble
+Mix
+Image
+SegZScribbleSeg
+7
+and computation of prior, including label class proportion prior, spatial prior
+and shape prior. (3) integration to develop deep neural network (referred to as
+ZScribbleNet) having losses of partial cross entropy (Lpce), global consistency
+(Lglobal), spatial prior loss (Lspatial), shape regularization (Lshape) and training
+strategy of supervision augmentation and prior regularization. Figure 1 presents
+the roadmap of the proposed framework.
+2.2
+Principle and modeling of efficient scribbles
+We investigate the principles of efficient scribbles and derive the objective of
+maximizing supervision with minimal annotation efforts. This leads to the pro-
+posal of supervision augmentation. In addition, we propose a global consistency
+loss to penalize the non-equivalence in the augmentation.
+Principles of efficient scribbles We shall verify the two principles of achiev-
+ing efficient scribble annotation in terms of maximal supervision later through
+the experiments in Section 3.2:
+(1) The large proportion of pixels annotated by scribbles compared with the
+whole set.
+(2) The randomness of distribution of scribbles. This is represented by the ran-
+dom and wide-range annotations.
+Firstly, we are motivated by the knowledge that model training benefits from
+the finer gradient flow through larger proportion of annotated pixels [39]. There-
+fore, we try to increase the annotation proportion with the same effort. One
+natural idea is to simply expand the width of scribbles. However, this way only
+increases the label amount in local area, and lacks the ability to enlarge anno-
+tation range across the entire image.
+Secondly, we are inspired by the fact that the imaging data are easier to be
+restored from random samples of pixels than from down-sampled low-resolution
+images with regular patterns [13]. This was due to the fact that the randomly
+and sparsely distributed samples maintain the global structure of the imaging
+data, which therefore can be restored with existing low-rank or self-similarity
+regularization terms. By contrast, the regularly down-sampled low-resolution
+images have evidently reduced tensor ranks, compared with the original high-
+resolution data, thus lose the global structure information. Motivated by this, we
+assume the features of full segmentation (similarly to the global structure infor-
+mation) can be portrayed (restored) with sparse scribble annotations randomly
+and widely distributed within the entire dataset. With such scribble annotation,
+the segmentation network can easily learn the global shape prior.
+Based on the observations described above, we propose to model efficient
+scribbles by supervision augmentation simulating large annotation proportion
+and randomness of scribble distribution.
+Modeling via supervision augmentation We aim to generate training im-
+ages with efficient scribbles by maximizing the supervision via mixup operations
+
+8
+K Zhang & X Zhuang
+and achieving the randomness via occlusion operations. This resembles data
+augmentation, which increases the data diversity and enables robust training.
+Search optimal annotation with mixup: Motivated by the principles of ef-
+ficient scribble, we first seek the optimal scribble with large annotated ratio,
+high supervision, and the unchanged local features. To achieve that, instead of
+maximizing the annotations directly, we aim to maximize the saliency of mixed
+images, which measures the sensitivity of model to inputs. Given that the an-
+notated area tends to be accompanied with high saliency, maximizing saliency
+also increases the scribble annotations.
+For two image-scribble pairs (X1, Y1), (X2, Y2) of dimension n, we denote
+the resulted mixed image-label pair as (X′
+12, Y ′
+12). The transportation process is
+defined by:
+X′
+12 = T(X1, X2) and Y ′
+12 = T(Y1, Y2),
+(1)
+T(X1, X2) = (1 − β) ⊙ �
+1 X1 + β ⊙ �
+2 X2,
+(2)
+where T(X1, X2) represents the transportation process between image X1 and
+X2; �
+i denotes the transportation matrix of size n×n for image Xi; β means the
+mask with value [0, 1] of dimension n; ⊙ is the element-wise multiplication. Then,
+we aim to maximize the saliency of transportation result over the parameters
+{�
+1, �
+2, β}:
+{�
+1, �
+2, β} = arg max
+�
+1,�
+2,β
+[(1 − β) ⊙ �
+1M(X1) + β ⊙ �
+2M(X2)],
+(3)
+where M(X) denotes the saliency map of image X, which is obtained by com-
+puting the l2 norm of gradient values. We solve this optimization problem based
+on PuzzleMix [18]. To preserve the local statistic features, the optimization ob-
+jective also includes the image local smoothness, and the mixing weight prior.
+For details of the optimization objective, we refer readers to PuzzleMix [18] and
+Appendix A of supplementary materials.
+Introduce randomness via occlusion: We propose to simulate randomly
+distributed scribbles via occlusion. Specifically, one square area of the mixed
+image is randomly dropped and replaced with the background. Since that the
+proportion of the background annotated by scribbles tends to be smaller than
+that of the foreground classes, the occlusion operation alleviates the imbalance
+problem of class mixture ratios within labeled pixels, and further improves the
+results of mixture ratio estimation, which will be elaborated in Section 2.3.
+We denote the occluded image-label pair as (X′′, Y ′′), which is obtained by:
+X′′
+12 = (1 − 1b) ⊙ X′
+12
+(4)
+Y ′′
+12 = (1 − 1b) ⊙ Y ′
+12
+(5)
+where 1b denotes a rectangular mask of size n × n with value in [0, 1]. The
+rectangular mask is randomly rotated to occlude the mixed image, and turns
+
+ZScribbleSeg
+9
+Fig. 3. Illustration of supervision augmentation and global consistency. Supervision
+maximization is achieved with the mix augmentation to increase the annotated pro-
+portion and data variety. Global consistency requires the segmentation result of mixed
+image and unmixed image to be consistent.
+the occluded area into background. Following [49], we set the size of rectangular
+to be 32 × 32.
+Global consistency loss: The objective of global consistency regularization is
+to leverage the mix-invariant property. As Figure 3 shows, global consistency
+requires the same image patch to have consistent segmentation in two scenarios,
+i.e., the unmixed image and the mixed image. Let the segmentation result of
+image X predicted by network be ˆY = f(X). For the transported image X′
+12 =
+T(X1, X2), the consistency of mixup is formulated as:
+T(f(X1), f(X2)) = f(T(X1, X2)),
+(6)
+which requires the segmentation of mixed image to be consistent with the mixed
+segmentation, after the same transportation process. When applying the occlu-
+sion operation, we further have:
+(1 − 1b) ⊙ T( ˆY1, ˆY2) = f ((1 − 1b) ⊙ T(X1, X2)) .
+(7)
+Then, we propose to minimize the distance between two sides of Eq.(7). Let
+u12 = (1 − 1b) ⊙ T( ˆY1, ˆY2) and v12 = f ((1 − 1b) ⊙ T(X1, X2)). The negative
+cosine similarity Ln(u12, v12) is defined as:
+Ln(u12, v12) = −
+u · v
+||u12||2 · ||v12||2
+.
+(8)
+Taking the symmetrical metric into consideration, we similarly penalize the in-
+consistency between u21 and v21. Therefore, the global consistency loss is for-
+mulated as:
+Lglobal = 1
+2 [Ln(u12, v12) + Ln(u21, v21)] .
+(9)
+
+Supervision
+Global
+Image 1
+Seg 1
+Scribble 1
+augmentation
+consistency
+pce
+Seg mixed
+Mixed scribble
+Mixed seg
+Mix
+Occlusion
+Lglobal
+Network
+(Eq.9)
+pce
+Seg 2
+Scribble 2
+Image 2
+pce10
+K Zhang & X Zhuang
+Fig. 4. Illustration of spatial prior loss (Lspatial) for correction of prediction, via class
+mixture ratios (π) and spatial prior (with spatial energy).
+Discussion: Mixup operations could change the shape of target structures, re-
+sulting in the unrealistic image. To tackle it, as shown in Figure 3, we propose
+to combine the partial cross entropy (PCE) loss for labeled pixels of both mixed
+and unmixed image, and leverage mix equivalence to preserve shape consistency
+at global level. To further exploit the shape features, we propose to correct the
+network prediction guided by computed prior, which is described in Section 2.3.
+2.3
+Modeling and computation of prior
+As shown in Figure 1, we model class mixture ratios, spatial prior, and shape
+prior to better capture global shape information and regularize the network
+training. As visualized in Figure 4, we compute the spatial energy to reflect the
+probabilities of pixels belonging to each class. We propose a new formulation to
+estimate critical prior of label class proportions, referred to as π, which guides
+the correction of erroneous network prediction.
+Problems statement The segmentation network trained with scribbles tends
+to generate under segmentation results of the target structures. Considering that
+the annotated ratio of classes can be imbalanced, the scribble supervised learning
+also brings challenges to the estimation of class mixture ratios π.
+Under segmentation: As shown in Figure 5, under segmentation refers to the
+results, where the size of segmented structure is generally smaller than ground
+truth, a phenomenon caused by the imbalanced annotated proportion and missed
+shape information. To solve the problem, we propose to evaluate π and spatial
+prior, which are crucial for the shape refinement. The accurate estimation of
+π can correct the imbalanced label ratios, and enable model to adjust the size
+of segmentation result. The computation of spatial prior is able to encode the
+feature similarity between pixels, and rectify the shape of target structures. We
+
+Correction of prediction
+Scribble
+Left ventricle
+Spatial energy
+T estimation
+<-
+Prediction
+Under segmentation
+Corrected shape
+Spatial priors
+Class mixture ratios
+Right ventricle
+Adjusted prediction
+4
+Lspatial
+(Eq.29)ZScribbleSeg
+11
+(a)
+(b)
+Fig. 5. Two examples of under segmentation, pointed by the red arrows: (a) under
+segmented foreground labels from ACDC segmentation, i.e., left ventricle and right
+ventricle; (b) under segmented background from MyoPS segmentation.
+encode π and spatial prior with spatial prior loss, by ranking the spatial energy
+and select the top π ratio as the segmentation. To estimate π, we start from the
+imbalanced annotated ratios (referred to as a) and adapt it from labeled pixels
+to unlabeled pixels.
+Note that the problem of under segmentation can be even worse without
+the modeling of efficient scribbles. In the case of manually annotated scribbles,
+the resulting annotations may be distributed in a non-random pattern due to
+fixed labeling habits, resulting in the biased label distribution across the whole
+dataset. This problem could be alleviated by simulating randomly distributed
+labels through our proposed supervision augmentation.
+Challenges of π estimation: The evaluation of class mixture ratios is a criti-
+cal bottleneck in semi-/ weak-/ non-supervised learning, and serves as the basis
+of classes identification [14] and variance reduction [47,38]. However, existing
+methods are mainly proposed for binary classification, and can not be adapted
+to multi-class scenario directly. For segmentation task, the class mixture ratios
+are both imbalanced and interdependent, leading to the decrease in the perfor-
+mance of previous binary estimation approaches. Despite the class imbalance
+problem, the scribble supervised segmentation is also faced with the imbalance
+of annotated class ratios. For example, the annotated ratio of the background
+tends to be much smaller than that of the foreground classes. The imbalance of
+annotated ratio further enhances the difficulty of π estimation.
+Estimation of class mixture ratios π To tackle the under segmentation, we
+propose to estimate the class mixture ratios within unlabeled pixels.
+Objective: We aim to determine π to maximize the likelihood of observed
+unlabeled pixels. For nu unlabeled pixels x = [x1, x2, · · · , xnu] sampled from
+
+12
+K Zhang & X Zhuang
+pu(x), the likelihood of these unlabeled pixels is formulated as:
+L(π) =
+nu
+�
+i=1
+pu(xi) =
+nu
+�
+i=1
+[
+m
+�
+k=1
+pu(xi|ck)pu(ck)],
+(10)
+where pu(xi|ck) represents the within-class probability of class ck ∈ {c0, · · · , cm}
+for unlabeled pixel xi. We assume the within-class probabilities of labeled and un-
+labeled pixels to be unchanged. Then, we estimate π = [pu(c1), pu(c2), · · · , pu(cm)]
+to maximize the likelihood of unlabeled observations in Eq.( 10).
+To maximize the likelihood in Eq.(10), we follow the EM algorithm in [24,30]
+and introduce the unknown variable s = (s1, s2, · · · , snu), where si is the one-
+hot vector of dimension m with the i-th value equals 1. Then, the likelihood
+L(π|x, s) is written as:
+L(π|x, s) =
+nu
+�
+i=1
+m
+�
+k=1
+[pu(xi|ck)pu(ck)]sik .
+(11)
+The log likelihood l(π|x, s) is derived as:
+l(π|x, s) =
+nu
+�
+i=1
+m
+�
+k=1
+sik log(pu(xi|ck))
++
+nu
+�
+i=1
+m
+�
+k=1
+sik log(pu(ck))
+(12)
+E-step: The E-step of EM algorithm computes the expected value of l(s|x, π)
+given the observations x and current estimate of π[t],
+Q(π|x, π[t]) =E
+�
+l(π|s, x)|x, π[t]�
+=
+nu
+�
+i=1
+m
+�
+k=1
+E(sik|xi, π[t]
+k ) log(pu(xi|ck))
++
+nu
+�
+i=1
+m
+�
+k=1
+E(sik|xi, π[t]
+k ) log(pu(ck)),
+(13)
+where E(sik|xi, π[t]
+k ) is represented as:
+E(sik|xi, π[t]
+k ) = p(sik = 1|xi, π[t]
+k ) = p[t]
+u (ck|xi)
+(14)
+Estimation of p[t]
+u (ck|xi): To solve the current estimate of p[t]
+u (ck|xi), we aim
+to adapt the posteriori probability from labeled pixels to unlabeled pixels. For
+labeled pixels, the posteriori probability pl(ck|xi) is estimated by the model
+prediction. For class ck and pixel xi, Based on our assumption that the within-
+class probabilities of labeled and unlabeled pixels are same, we have
+pu(xi|ck) = pl(xi|ck),
+(15)
+
+ZScribbleSeg
+13
+Based on Bayes’ theorem, the within-class probabilities of labeled pixel pl(xi|ck)
+and unlabeled pixel pu(xi|ck) are written as:
+ˆpl(xi|ck) = ˆpl(ck|xi)p(xi)
+ˆpl(ck)
+(16)
+ˆpu(xi|ck) = ˆpu(ck|xi)ˆpu(xi)
+ˆpu(ck)
+(17)
+By substituting ˆpu(xi|ck) in Eq.(17) and ˆpl(xi|ck) in Eq.(16) into Eq.(15), we
+adapt the within-class probabilities from labeled pixels to unlabeled pixels as
+follows:
+ˆpu(ck|xi) = ˆpl(xi)
+ˆpu(xi) · ˆpu(ck)
+ˆpl(ck) ˆpl(ck|xi).
+(18)
+For binary estimation, the mixture ratio is independently estimated for each
+class, which does not leverage the inter-relationship between classes. For multi-
+class segmentation, we naturally utilize the condition that the sum of the prob-
+abilities of all classes equals to 1, i.e.,
+m
+�
+k=0
+ˆpu(ck|xi) = 1.
+(19)
+By combing Eq.(18) and Eq.(19), one can obtain:
+1 = ˆpl(xi)
+ˆpu(xi)
+m
+�
+k=0
+ˆpu(ck)
+ˆpl(ck) ˆpl(ck|xi).
+(20)
+Then, ˆpl(xi)/ˆpu(xi) is represented as:
+ˆpl(xi)
+ˆpu(xi) =
+� m
+�
+k=0
+[ˆpu(ck)ˆpl(ck|xi)/ˆpl(ck)]
+�−1
+.
+(21)
+By substituting ˆpl(xi)/ˆpu(xi) into Eq. (18), we can obtain the formulation of
+ˆpu(ck|xi) as follows:
+ˆpu(ck|xi) =
+ˆpu(ck)ˆpl(ck|xi)/ˆpl(ck)
+�m
+k=0[ˆpu(ck)ˆpl(ck|xi)/ˆpl(ck)].
+(22)
+Therefore, the current estimate of posteriori probability ˆpu(ck|xi) is written
+as:
+ˆpt
+u(ck|xi) =
+πt
+k ˆpl(ck|xi)/ˆpl(ck)
+�m
+k=0[πt
+k ˆpl(ck|xi)/ˆpl(ck)],
+(23)
+where ˆpl(ck) is empirically evaluated by the class frequency within labeled pixels,
+i.e., ˆpl(ck) = nk
+l /nl.
+M-step: The M-step maximizes Q(π, π[t]) in Eq.(13), i.e.,
+π[t+1] := arg max
+π
+Q(π|x, π[t])
+(24)
+
+14
+K Zhang & X Zhuang
+We empirically solve the πt+1
+k
+as:
+π[t+1]
+k
+= 1
+nu
+nu
+�
+i=1
+p[t]
+u (ck|xi)
+(25)
+The π[t]
+k is initialized with the class frequency within labeled pixels a, with
+ak = nk
+l
+nl . Then, the E-step of Eq.(13) and M-step of Eq.(25) is repeated until
+the estimation of π converges. The posteriori probability ˆpu(ck|xi) and priori
+probability ˆpu(ck) are re-estimated in each iteration.
+Discussion: There are two conditions of the proposed algorithm. Firstly, we
+assume the within-class probabilities of labeled and unlabeled pixels be the same,
+which means the labeled pixels should be randomly sampled based on classes.
+Secondly, π is initiated with the class frequency of labeled pixels a. Since that
+the annotated ratio of background is smaller than that of the foreground classes,
+the priori probabilities of foreground classes within unlabeled pixels tend to
+be over-estimated. The first problem can be tackled by modeling the efficient
+scribbles, to achieve the random distribution of annotations. For the second
+problem, by randomly occluding the image and replace the occluded area with
+background, we are able to increase the ratio of background and alleviate this
+problem to some extent. Furthermore, we propose to address it with the marginal
+probability maximization, which will be explained in Section 2.3.
+Computation of spatial energy Given the estimated class mixture ratios,
+we aim to identify the unlabeled pixels by determining the probability of pixels
+belonging to each class. Instead of using model predictions directly, we further
+encode the spatial relationship to compensate the inaccurate results generated
+by segmentation network. Inspired by [31], we estimate the spatial energy of
+unlabeled pixels with energy term in a dense setting.
+Firstly, we use Gaussian kernels Gij to measure the distance between pixels
+at position i and j as:
+Gij = exp
+�
+−(pi − pj)2
+2σ2p
+− (oi − oj)2
+2σ2o
+�
+,
+(26)
+where pi represents the position of pixel xi; oi denotes the color feature; σp and
+σo are the bandwidth parameters for position and color information, respectively.
+The shallow features like color and position are specific to the pixel and do not
+rely on the network prediction. Then, the energy term φij leveraging prediction
+ˆy is formulated as:
+φij(ˆy) = Gij ˆyiˆyj,
+(27)
+which denotes the pairwise relationship between two pixels. This energy term
+connects every pixels with each other within one image. Based on φi,j, we define
+the element of spatial energy Φ in a dense setting, i.e.,
+Φi(ˆy) =
+�
+j∈Ωi
+φij(ˆy),
+(28)
+
+ZScribbleSeg
+15
+where Ωi = {Pos(i) − Pos(j) ≤ r}, means the neighborhood window of radius r.
+Instead of taking the total energy as the regularization loss as [31], we consider
+Φ as the spatial energy to reflect the relative probability of pixels belonging to
+each class.
+Spatial prior and shape prior losses Spatial prior loss is computed by
+ranking the spatial energy and selecting the top π proportion of pixels as the
+segmentation. Considering that adjusting multiple structures directly can be
+challenging, we instead separate each foreground class from the others, and
+then tackle the individual structure. Given that the mixture ratios of foreground
+classes tend to be over-estimated, we instead leverage the accurate negative pix-
+els filtered by estimated mixture ratios, and maximize the marginal probability
+of these pixels belonging to other classes.
+Firstly, by ranking the spatial energy and applying the mixture ratio of each
+class, we are able to distinguish negative pixels from unlabeled pixels. For fore-
+ground class ck, we rank the unlabeled pixels according to the spatial energy Φk
+of class ck in Eq. (28). Given the estimated mixture ratio πk, we set pixels in
+the top πk proportion to be positive samples Ωk Correspondingly, the remaining
+pixels are taken as negative pixels, denoted as ¯Ωk. Taking over-estimated πk into
+account, we believe the set of negative pixels ¯Ωk is more accurate than Ωk.
+Secondly, we design the spatial prior loss (Lspatial) based on maximal marginal
+probability of negative samples ¯Ωk belonging to other classes. For each class
+ck, we take it as foreground and fuse other classes except ck into background.
+The fused class is denoted as ¯ck. For pixel xi in ¯Ωk, its marginal probabil-
+ity belonging to ¯ck equals the sum of probabilities of the fused classes, i.e.,
+ˆp(¯ck|xi, xi ∈ ¯Ωk) = �m
+k′=1[1[k′̸=k]ˆp(ck|xi)]. To maximize the marginal proba-
+bility of negative pixel xi belonging to ¯ck, we formulate the spatial prior loss
+as:
+Lspatial = −
+m
+�
+k=1
+�
+xi∈ ¯
+Ωk
+log(ˆp (¯ck|xi)) .
+(29)
+Shape prior loss is developed to regularize inter-connected structures in the
+segmentation results. This loss is adopted to further reduce noise and smooth
+boundary. It requires the model prediction to be consistent with its maximum
+connected area, and minimizes their cross entropy loss, i.e.,
+Lshape = −
+�
+k∈Ψ
+F( ˆYk) log( ˆYk),
+(30)
+where Ψ is the set of label classes with inter-connected structures; F(·) denotes
+the morphological function, and outputs the largest inter-connected area of input
+label.
+2.4
+ZScribbleNet
+ZScribbleSeg is achieved via a deep neural network referred to as ZScribbleNet.
+ZScribbleNet does not depend on any particular network architecture, and can
+
+16
+K Zhang & X Zhuang
+Table 1. Efficiency analysis of scribble forms for regular structure segmentation of
+cardiac ventricles (ACDC dataset) and irregular segmentation of myocardial pathology
+(MyoPS dataset). Here, Nscribble and Npix respectively denote the number of manual
+draws to generate scribble annotations and number of annotated pixels, which indicate
+annotation efforts; k is the number of manual draws (scribbles) and n is the given
+threshold of annotation efforts, where k << n. Segmentation results are evaluated on
+test set and reported in Dice scores.
+Methods
+Nscribble Npix
+Structural segmentation
+Irregular segmentation
+LV
+MYO
+RV
+Avg
+Scar
+Edema
+Avg
+Points
+n
+n
+.876±.134 .801±.089 .858±.081 .845±.107 .551±.246 .638±.115 .595±.194
+Skeleton
+k
+n
+.805±.145
+.737±.095
+.769±.128
+.770±.126
+.504±.213
+.057±.022
+.281±.271
+Random walk
+k
+n
+.798±.173
+.698±.153
+.753±.157
+.744±.165
+.516±.284
+.529±.123
+.522±.184
+DirRandomWork
+k
+n
+.844±.143
+.755±.102
+.798±.173
+.799±.146
+.539±.217
+.637±.108
+.588±.176
+be directly applied to any CNN backbone. For all experiments, we adopt the
+variant of UNet [1] as the backbone of segmentation network. As Figure 2 shows,
+two images are mixed together to perform the supervision augmentation. Then,
+our ZScribbleNet takes the mixed images and unmixed images as the input, and
+output their segmentation results.
+For model training, images and their scribble annotations are sampled to
+estimate the training objective (L), which is formulated as:
+L = Lpce + λ1Lglobal + λ2Lspatial + λ3Lshape
+�
+��
+�
+unsup
+,
+(31)
+where Lpce is the partial cross entropy loss calculated for annotated pixels in
+unmixed image and mixed image; the global consistency loss Lglobal in Eq.(9)
+requires the mix equivalence for the supervision augmentation; spatial prior loss
+Lspatial in Eq.(29) encodes the π prior and spatial prior; shape regularization
+loss Lshape in Eq.(30) leverages shape prior; λ1, λ2, λ3 are hyper-parameters to
+leverage the relative importance of different loss components.
+In the training phase, We warmly started training the networks with partial
+cross entropy loss Lpce, global consistency loss Lglobal, and shape regularization
+loss Lshape for 100 epochs, and then invoked the spatial loss Lspatial. In the
+testing phase, the trained network predicted the segmentation results of input
+image directly.
+3
+Experiments and Results
+We first investigated a variety of scribble forms, and analyzed the principles
+of efficient scribbles in Section 3.2. Then, we performed ablation study to the
+proposed ZScribbleSeg in Section 3.3. Finally, we demonstrated the performance
+of ZScribbleSeg with comparisons to other state-of-the-art methods in various
+segmentation tasks using four open datasets in Section 3.4.
+
+ZScribbleSeg
+17
+(a)
+(b)
+(c)
+(d)
+Fig. 6. Performance of segmentation networks trained by the Points scribble form
+with different number of pixels Npix, with comparisons to fully supervised models
+(FullSupUNet): (a) and (c) visualize Dice scores with respect to different Npix on ACDC
+and MyoPS, respectively. The performance of models trained by the Random walk
+form, with increasing step length l, compared with models trained by DirRandWalk:
+(b) and (d) show the Dice scores of segmentation on ACDC and MyoPS, respectively,
+given Npix = n.
+3.1
+Materials
+Tasks and datasets Our validation included four segmentation tasks, including
+(1) regular structure segmentation of cardiac ventricles from anatomical imag-
+ing using ACDC dataset, (2) regular structure segmentation from pathology en-
+hanced imaging with a smaller training size using MSCMRseg dataset, (3) irreg-
+ular object segmentation of myocardial pathology from multi-modality imaging
+using MyoPS dataset, and human pose segmentation from natural scene images
+using PPSS dataset.
+ACDC dataset was from the MICCAI’17 Automatic Cardiac Diagnosis
+Challenge [4]. This dataset consists of short-axis cardiac images using anatomi-
+cal MRI sequence (BSSFP) from 100 patients, with gold standard segmentation
+of cardiac ventricular structures, including left ventricle blood cavity (LV), left
+
+0.85
+0.84
+0.83
+Dice
+0.82
+0.81
+Points
+FullSupUNet
+0.80
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+Number of annotated pixels on ACDC(n)0.81
+0.80
+0.79
+Score
+0.78
+Dice
+0.77
+0.76
+Random Walk
+0.75
+DirRandomWalk
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Stepsizeofrandomwalkon ACDC0.63
+0.62
+Score
+0.61
+Dice
+0.60
+0.59
+0.58
+Points
+0.57
+FullSupUNet
+1
+2
+3
+4
+5
+Number of annotated pixels on MyoPS(n)0.59
+0.58
+0.57
+Score
+0.56
+Dice
+0.55
+0.54
+0.53
+RandomWalk
+DirRandomWalk
+0.52
+1
+2
+3
+4
+5
+6
+7
+8
+StepsizeofrandomwalkonMvoPs18
+K Zhang & X Zhuang
+ventricle myocardium (MYO), and right ventricle blood cavity (RV). For exper-
+iments, we randomly divided the 100 subjects into a training set of 70 subjects,
+a validation set of 15 subjects (particularly for ablation study), and a test set of
+15 subjects.
+MSCMRseg was from the MICCAI’19 Multi-sequence Cardiac MR Seg-
+mentation Challenge [59,58], consisting of images from 45 patients with car-
+diomyopathy and the gold standard segmentation of LV, MYO and RV. We
+employed the 45 images of late gadolinium enhanced (LGE) MRI to evaluate
+the segmentation of ventricle structures. Following [48], we divided the 45 im-
+ages into three sets of 25 (training), 5 (validation), and 15 (test) images for
+all experiments. Note that this structure segmentation is more challenging than
+that on ACDC due to its smaller training set and pathology enhanced images.
+MyoPS dataset was from MICCAI’20 Myocardial pathology segmentation
+Challenge [26], consisting of paired images of BSSFP, LGE and T2 cardiac MRI
+from 45 patients. The task was to segment the myocardial pathologies, includ-
+ing scar and edema, which do not have regular shape or structure thus their
+segmentation represents a different task to the regular structure segmentation.
+Following the benchmark study [26], we split the data into 20 pairs of training
+set, 5 pairs of validation set and 20 pairs of test set.
+PPSS refers to the Pedestrian Parsing on Surveillance Scenes (PPSS) dataset [28].
+We employed the task of human pose segmentation to validate the generaliz-
+ability of models on natural scene images. PPSS is a large scale human pars-
+ing dataset including 3673 annotated samples of 171 surveillance videos. The
+ground truth segmentation of eight classes including hair, face, upper clothes,
+arms, lower clothes, legs, shoes, and background were provided. We used the
+first 100 surveillance scenes for training and the remaining 71 videos for test.
+Evaluation metrics For experiments on ACDC, MSCMRseg and MyoPS datasets,
+we reported the Dice score and Hausdorff Distance (HD) on each foreground
+class separately following the practice of medical image segmentation. On PPSS
+dataset, we measured the multi-class Dice scores following [42], where Dice=
+2|ˆyy|
+|ˆy|+|y|, and ˆy and y denote the multi-channel prediction and ground truth la-
+bel, respectively.
+Pre-processing and implementation The two dimensional slices from ACDC
+and MSCMR datasets were of different resolutions. Hence, we first re-sampled all
+images into a fixed resolution of 1.37 × 1.37 mm and then extracted the central
+patch of size 212 × 212 for experiments. For MyoPS, we took the paired slices of
+BSSFP, LGE, and T2 CMR and cropped their center patches of size 192 × 192
+for experiments. We normalized the intensity of these medical images to be zero
+mean and unit variance. For PPSS dataset, we first re-sampled all images into
+the same resolution, and then padded the images to the size of 160 × 160. The
+intensities of images were normalized to a range between 0 and 1.
+For random occlusion, a square area of 32 × 32 was randomly occluded for
+each image. For the estimation of spatial energy, We adopted Gaussian kernels
+
+ZScribbleSeg
+19
+with color bandwidth σo = 0.1, position bandwidth σp = 6, and kernel radius
+r = 5. The hyper-parameters λ1, λ2, λ3 in Eq. (31) were empirically set to be
+0.05, 1, and 1, respectively.
+All models were trained with a batch size of 4, learning rate of 1e−4, and
+augmentation of flipping and random rotation. We implemented our models
+with Pytorch. All models were trained on one NVIDIA 3090Ti 24GB GPU for
+1000 epochs.
+Table 2. Results in Dice scores and Hausdorff Distance (HD) of the ablation study
+using ACDC dataset, where the models were evaluated on the validation set. Note that
+model #6 is ZScribbleSeg. Bold denotes the best result, and underline indicates the
+best but one in each category.
+Results in Dice
+Lpce Efficiency Lshape Lglobal Lspatial
+LV
+MYO
+RV
+Avg
+model #1
+✓
+×
+×
+×
+×
+.863±.089
+.804±.063
+.774±.150
+.813±.112
+model #2
+✓
+✓
+×
+×
+×
+.870±.100
+.833±.063
+.843±.076
+.848±.082
+model #3
+✓
+×
+✓
+×
+×
+.915±.068
+.871±.056
+.871±.058
+.886±.064
+model #4
+✓
+✓
+×
+✓
+×
+.920±.064
+.868±.051
+.886±.051
+.891±.059
+model #5
+✓
+×
+×
+×
+✓
+.923±.078
+.869±.051
+.889±.056
+.894±.066
+model #6
+✓
+✓
+✓
+✓
+✓
+.929±.057
+.876±.051
+.892±.049
+.899±.056
+Results in HD (mm) Lpce Efficiency Lshape Lglobal Lspatial
+LV
+MYO
+RV
+Avg
+model #1
+✓
+×
+×
+×
+×
+81.86±40.40
+65.97±33.62 60.91±44.62 69.58±40.37
+model #2
+✓
+✓
+×
+×
+×
+119.78±19.14 23.90±17.32 52.38±23.40 65.35±45.06
+model #3
+✓
+×
+✓
+×
+×
+4.45±5.39
+15.24±23.90 25.78±22.44 15.16±20.89
+model #4
+✓
+✓
+×
+✓
+×
+12.12±18.26
+29.41±24.56 16.97±15.62 19.50±20.94
+model #5
+✓
+×
+×
+×
+✓
+28.95±36.57
+44.77±34.69
+7.51±5.34 27.08±32.76
+model #6
+✓
+✓
+✓
+✓
+✓
+6.09±8.53
+11.14±14.53
+8.86±5.88
+8.70±10.40
+3.2
+Efficiency of scribble forms
+In this study, we first compared four scribble forms to illustrate the efficacy of
+randomly annotated scribbles for supervision. Denoting the number of annotated
+pixels using a manual and skeleton-wise scribble form as n, we generated other
+scribble forms with the same annotated ratios for a fair comparison. Then, we
+studied the performance of segmentation with respect to the number of pixels
+annotated using a random and wide range scribble form, by setting the number
+of annotated pixels to different times of n. Finally, we further explored variants
+of random walk annotations to show the importance of wide range in the random
+distribution of scribbles.
+We applied two segmentation tasks, i.e., regular structure segmentation of
+the cardiac ventricles on ACDC dataset and irregular segmentation of myocardial
+pathologies using MyoPS dataset. To compare the supervision of scribble forms
+directly, we trained all models with partial cross entropy (PCE) loss calculated
+for annotated pixels from scribbles. All experiment results were reported on the
+test set.
+Scribble forms One can measure the efforts of scribble annotations from two
+perspectives, i.e., number of manual draws to generate scribble annotations
+
+20
+K Zhang & X Zhuang
+Table 3. Results and comparisons of regular structure segmentation on ACDC dataset.
+These models were evaluated on the test set.
+Methods
+Dice
+HD (mm)
+LV
+MYO
+RV
+Avg
+LV
+MYO
+RV
+Avg
+PCE
+.805±.145
+.737±.095
+.769±.128
+.770±.126 62.55±36.04 68.30±27.77 59.62±42.62
+63.40±35.76
+WSL4 [29]
+.835±.164 .825±.032 .787±.191
+.792±.166 16.48±16.01 24.48±22.74 18.21±11.30
+19.72±17.67
+GatedCRF [31] .846±.157
+.744±.108
+.822±.111
+.804±.135 37.38±46.37 22.30±15.72 20.88±11.85
+26.85±30.03
+MAAG [42]
+.879
+.817
+.752
+.816
+25.23
+26.83
+22.73
+24.93
+CVIR [14]
+.866±.127
+.797±.102
+.737±.130
+.800±.130 47.51±50.82 10.70±8.39
+14.39±9.00
+.24.20±34.17
+nnPU [20]
+.862±.134
+.792±.124
+.829±.102
+.828±.123 67.28±48.60 18.60±17.93
+14.64±8.39
+33.51±38.43
+CycleMix [52]
+.876±.096
+.794±.083
+.829±.099
+.833±.098 16.60±19.90 18.04±17.78 19.09±21.44
+17.91±19.57
+ShapePU [53]
+.885±.103
+.806±.096
+.851±.089
+.848±.100 20.17±22.40 41.81±33.40 20.06±26.43
+27.35±29.33
+ZScribbleSeg
+.900±.065 .825±.069 .862±.102 .862±.086 7.69±6.94 8.93±6.40 12.74±12.48 9.79±9.19
+FullSupUNet
+.882±.123
+.824±.099
+.856±.112
+.854±.113 11.94±13.58 12.65±12.52
+14.82±9.69
+13.14±11.97
+(Nscribble) and number of annotated pixels (Npix). Given the certain amount
+of efforts, we designed four forms following different generation procedures, i.e.,
+(1) Skeleton, (2) Random walk, (3) Directed random walk (DirRandomWalk),
+(4) Points, and compared the segmentation performance of models trained us-
+ing such scribble annotations for supervision. The details of scribble forms are
+described bellow.
+Skeleton indicates the widely adopted scribble form by a rater, who approx-
+imately outlines the shape of each label class within the segmentation mask. For
+a segmentation task with k label classes, including the background, one needs k
+manual draws (scribbles) for a training image. For ACDC dataset, we adopted
+the manual annotated skeleton scribble released by [42]; while for pathologies
+in MyoPS dataset, we generated the skeleton scribbles automatically using the
+skeletonization algorithm [36]. We refer the reader to Appendix B of the supple-
+mentary material for generation details.
+Random walk starts from a random point within the segmentation mask.
+Then, the annotation moves along a random direction of image lattice within the
+segmentation mask, with a given step length (l by default set to 1). We repeated
+such moves until the ratio or number of annotated pixels reached a threshold
+(n).
+Directed random walk, DirRandomWork for short, is the random walk
+with momentum. The scribble generated by Random walk tends to cluster within
+a local area of the radius √r given r-step walks. To achieve wide range distri-
+bution without manually setting the step length (l), we therefore adopted this
+directed random walk, which prefers moving along the same direction to the pre-
+vious step. If the next point does not lie in the segmentation mask, we changed
+the walking direction to be along the smallest angle to the previous one.
+Points scribble form refers to an ideal form, which randomly samples anno-
+tated pixels within the segmentation mask. However, it is difficult to generate
+such scribble annotation in practice, due to the huge number of manual draws
+which equals the number of annotated pixels, i.e., Nscribble = Npix. Therefore,
+we considered this form as the upper bound of scribble supervision under the
+same ratio of annotated pixels.
+
+ZScribbleSeg
+21
+Image
+Ground Truth
+PCE
+CVIR
+nnPU
+WSL4
+GatedCRF
+CycleMix
+ShapePU
+ZScribbleSeg      FullSupUNet
+Dice (Avg)
+:MYO
+:RV
+Median
+Worst
+:LV
+0.758
+0.827
+0.894
+0.852
+0.870
+0.897
+0.902
+0.907
+0.903
+0.486
+0.390
+0.472
+0.628
+0.618
+0.386
+0.262
+0.773
+0.544
+Dice (Avg)
+Scribble
+Fig. 7. Visualization of cardiac segmentation on ACDC dataset. The two slices were
+from the median and the worst cases by the average Dice scores of all compared meth-
+ods.
+Results Given the same amount of annotated pixels, we show the effect of dif-
+ferent scribble forms on regular structures (ACDC) and irregular objects (My-
+oPS). As Table 1 illustrates, when the four scribble forms had the same number
+of annotated pixels Npix, Points achieved the best Dice scores on both of the
+structural segmentation and irregular segmentation tasks, thanks to the effects of
+randomness and wide range distribution of scribbles. However, when we limited
+the efforts of manual draws to be the same, DirRandomWalk became more favor-
+able, as the scribble form of Points could be impractical. Furthermore, Skeleton
+scribble was illustrated to be the least efficient form, particularly the segmenta-
+tion network trained on such dataset performed poorly on the irregular object
+segmentation task. This was probably due to the fact that when the target was
+difficult to outline, Skeleton form could fail to portray the entire segmentation,
+leading to poor performance or even a failure in training the segmentation net-
+works. On the contrary, randomly distributed scribble forms, such as Random
+walk and DirRandomWalk, demonstrated their superiority, particularly on the
+irregular object segmentation with remarkable improvements on average Dice
+over Skeleton of 24.1% and 30.7%, respectively.
+Number of annotated points: By varying the number of annotated pixels
+(Npix), we validated the influence of annotated proportions on scribble super-
+vised segmentation. As shown in Figure 6 (a) and (c), the model performance
+tended to be improved as Npix increases, indicating that model training bene-
+fited from larger proportion of annotated pixels. One can observe from Figure 6
+(a) that the segmentation performance started converging when Npix reached
+2n. By contrast, for the more difficult segmentation task on irregular objects, as
+Figure 6 (c) illustrates, the model performance converged after Npix ≥ 4n.
+Wide-ranged distribution: We further investigated the influence of wide
+range distribution of scribbles, by training networks with varying step length l
+in Random walk. As the step length increases, the label distribution range of
+Random walk gradually expanded. From Figure 6 (b) and (d), one can see that
+the segmentation performance of average Dice scores was improved as the step
+length increased, and the performance gradually converged to that of DirRan-
+domWalk. This confirmed that the widely distributed scribbles were better to
+provide finer supervision under the same number of draws and annotated pixels.
+
+22
+K Zhang & X Zhuang
+Table 4. Results and comparisons of regular structure segmentation on pathology
+enhanced images (LGE CMR) using MSCMRseg dataset.
+Methods
+Dice
+HD (mm)
+LV
+MYO
+RV
+Avg
+LV
+MYO
+RV
+Avg
+PCE
+.514±.078
+.582±.067
+.058±.023
+.385±.243
+259.4±14.19
+228.1±21.36
+257.4±12.43
+248.3±21.63
+WSL4 [29]
+.902±.040
+.815±.033
+.828±.101
+.848±.076
+55.95±4.88
+42.07±13.48
+32.08±6.57
+43.37±31.04
+GatedCRF [31] .917±.044
+.825±.032
+.848±.073
+.863±.066
+25.72±4.37
+37.92±5.10
+32.83±5.59
+32.16±7.11
+CVIR [14]
+.331±.076
+.371±.088
+.404±.110
+.368±.095
+259.2±14.23
+243.0±13.76
+180.9±55.44
+227.7±47.63
+nnPU [20]
+.341±.067
+.538±.081
+.432±.100
+.437±.115
+259.4±14.19
+201.6±66.98
+199.7±57.50
+220.2±57.70
+CycleMix [52]
+.748±.064
+.730±.047
+.835±.041
+.771±.069 224.59±35.27 28.26±20.77
+73.36±51.39 108.74±92.65
+ShapePU [53]
+.880±.046
+.785±.080
+.833±.087
+.833±.082 178.02±50.93 178.05±25.39 189.35±55.78 181.81±45.27
+ZScribbleSeg
+.922±.039 .834±.039 .854±.055 .870±.058 12.10±14.70 16.52±19.14 51.03±39.27 26.55±31.39
+FullSupUNet
+.909±.049
+.821±.054
+.826±.087
+.852±.076
+10.02±12.36
+11.89±11.34
+56.91±41.99
+26.27±33.63
+3.3
+Ablation study
+We studied the effectiveness of the proposed strategies in modeling efficient scrib-
+bles and prior regularization for ZScribbleNet. We used the ACDC dataset and
+the expert-made scribble annotations released by [42], and evaluated the model
+performance on the validation set. We compared six ablated models which were
+trained with or without the usage of modeling efficient scribbles (denoted as
+Efficiency), and with different combinations of the four loss functions, i.e., the
+partial cross entropy (Lpce), the global consistency loss (Lglobal) in Eq.(9), the
+spatial prior loss (Lspatial) in Eq.(29), and the shape prior loss (Lshape) in Eq.(30).
+Table 2 presents the results. When model #2 adopted the proposed super-
+vision augmentation to model efficient scribbles (indicated by the column of
+Efficiency), its performance was improved compared to model #1, as one can
+see from their average Dice scores (0.848 vs. 0.813) and average HDs (65.35
+mm vs. 69.58 mm). This demonstrated the benefits of model training from the
+augmented supervision. When combining the supervision augmentation with the
+global consistency loss (Lglobal), leading to model #4, the performance was fur-
+ther boosted with remarkable improvements, namely 4.3% gain in Dice (0.891
+vs. 0.848) and over 45 mm error reduction in HD (19.50 mm vs. 65.35 mm). Al-
+ternatively, when leveraging inter connectivity via the shape regularization loss
+(Lshape), model #3 obtained an overwhelming improvement in HD, which was
+reduced from 69.58 mm to only 15.16 mm compared to model #1. This indicated
+the results were with much less noisy and outlier segmentation. We then further
+investigated the advantage of spatial prior (Lspatial) in training ZScribbleNet.
+One can see from the result of model #5 that it achieved the most evident gain
+in terms of Dice results, with an improvement of 8.1% (0.894 vs. 0.813) by solely
+including one extra loss. Finally, our ZScribbleSeg (model #6) achieved the best
+performance with average Dice of 0.899 and HD of 8.70 mm. This indicated that
+the combination of efficient scribbles and priors endowed the algorithm with sub-
+stantial supervision and prior knowledge for scribble-supervised segmentation.
+3.4
+Performance and Comparisons
+We conducted experiments over the four segmentation tasks stated in Sec-
+tion 3.1. (1) For the structural segmentation of cardiac ventricles from ACDC
+
+ZScribbleSeg
+23
+Image
+Ground Truth
+PCE
+CVIR
+nnPU
+ShapePU
+WSL4
+GatedCRF
+CycleMix
+ShapePU
+ZScribbleSeg 
+FullSupUNet
+Dice (Avg)
+:MYO
+:RV
+Median
+Worst
+:LV
+0.389
+0.353
+0.412
+0.886
+0.885
+0.787
+0.865
+0.893
+0.880
+0.370
+0.328
+0.428
+0.723
+0.814
+0.735
+0.723
+0.829
+0.830
+Scribble
+Dice (Avg)
+Fig. 8. Visualization of cardiac segmentation on LGE CMR using MSCMRseg dataset.
+The two slices were from the median and the worst cases by the average Dice scores of
+all compared methods.
+dataset, we used the expert-made scribbles released by [42]. (2) For the car-
+diac structural segmentation from pathology enhanced imaging (MSCMRseg)
+dataset, we used the manually annotated scribbles released by [52]. (3) For
+the irregular myocardial pathology segmentation from MyoPS dataset, we first
+adopted the standard skeletonization algorithm for the simulated scribble anno-
+tation of pathologies [36]. Then, we manually annotated skeleton scribbles for
+the structures of LV, Myo, RV and background. (4) For the human pose seg-
+mentation from PPSS dataset, we adopted the scribble annotations generated
+by the standard skeletonization algorithm [36].
+We compared ZScribbleSeg with eight to nine methods. We first implemented
+the PCE loss (Lpce) as a baseline method (referred to PCE). Then, we imple-
+mented four state-of-the-art (SOTA) scribble supervised segmentation methods,
+i.e., WSL4 [29], GatedCRF [31], CycleMix [52], and ShapePU [53] to run the
+same experiments. We cited the ACDC and PPSS results reported in [42] for the
+MAAG method, which is also a SOTA method for this task. Furthermore, we
+adopted two semi-supervised SOTA methods based on positive unlabeled learn-
+ing, i.e., CVIR [14] and nnPU [20], and re-implemented to adapt them for the
+scribble-supervised segmentation tasks. For more details of adaptation, the read-
+ers are referred to Appendix C of the supplementary material. Finally, we trained
+UNet with full annotations as a baseline of fully-supervised approach (referred
+to as FullSupUNet). Note that the post-processing steps of all experiments were
+removed for a fair comparison.
+Structure segmentation from anatomical images Table 3 presents the
+Dice and HD results of 10 approaches for regular structure segmentation of car-
+diac ventricles from ACDC dataset. One can observe that ZScribbleSeg achieved
+average Dice of 0.862 and HD of 9.79 mm, outperforming the other scribble-
+supervised methods evidently. The quantitative results of ZScribbleSeg were
+comparable to (or slightly better than) that of the fully supervised method (Full-
+supUNet) whose average Dice and HD are 0.854 and 13.14 mm, respectively.
+Particularly, the HD results of ZScribbleSeg (9.79 mm) and FullSupUNet
+(13.14 mm) were evidently much better than the other methods. Note that HD
+is highly sensitive to the noisy and outlier segmentation results, which are com-
+monly seen when the supervision of global shape information is not sufficient.
+
+24
+K Zhang & X Zhuang
+Table 5. Results and comparisons of irregular segmentation of myocardial pathologies
+on MyoPS dataset.
+Methods
+Dice
+HD (mm)
+Scar
+Edema
+Avg
+Scar
+Edema
+Avg
+PCE
+0.504±0.213
+0.057±0.022
+0.281±0.271
+82.68±33.95 147.61±20.59 115.15±43.00
+WSL4 [29]
+0.031±0.029
+0.106±0.033
+0.069±0.049 172.37±45.13 170.05±20.44 171.20±34.60
+GatedCRF [31] 0.020±0.013
+0.042±0.020
+0.031±0.019 173.60±44.98 170.10±20.44 171.8±34.53
+CVIR [14]
+0.505±0.214
+0.080±0.031
+0.293±0.263
+61.59±32.09 125.27±20.83 93.43±41.86
+nnPU [20]
+0.530±0.241
+0.085±0.035
+0.308±0.282
+48.88±23.55 125.27±20.83 87.07±44.47
+CycleMix [52]
+0.550±0.237
+0.626±0.124
+0.588±0.191
+65.64±42.81
+81.97±40.87
+73.81±42.13
+ShapePU [53]
+0.558±0.237
+0.615±0.144
+0.587±0.205
+57.33±31.58
+53.00±31.42
+55.16±31.17
+ZScribbleSeg
+0.596±0.237 0.676±0.113 0.636±0.188 46.73±20.04 47.05±24.30 46.89±21.98
+FullSupUNet
+0.607±0.253
+0.659±0.135
+0.633±0.202
+55.35±35.73
+63.53±33.15
+59.44±34.27
+Ground Truth
+PCE
+CVIR
+nnPU
+ShapePU
+CycleMix
+ZScribbleNet
+FullSupUNet
+Dice (Scar);
+Dice (Edema)
+0.488;
+0.039
+0.478;
+0.054
+0.667;
+0.062
+0.591;
+0.597
+0.558;
+0.616
+0.671;
+0.637
+0.716;
+0.713
+Median cases 
+:Scar
+:Edema
++
+(
+)
+Dice (Scar);
+Dice (Edema)
+0.563;
+0.042
+0.564;
+0.061
+0.677;
+0.059
+0.726;
+0.684
+0.707;
+0.698
+0.755;
+0.750
+0.705;
+0.686
+Image
+GatedCRF
+0.041;
+0.074
+0.028;
+0.056
+WSL4
+0.041;
+0.180
+0.028;
+0.101
+Scribble
+Fig. 9. Visualization of irregular segmentation of myocardial pathologies on MyoPS
+dataset. The two slices were from the median cases by average Dice scores of edema or
+scar segmentation of all compared methods.
+The results indicate the proposed efficient scribble modeling and prior regular-
+ization were able to alleviate the problem of inadequate supervision and incom-
+plete shape information from training images with scribble annotations. Finally,
+Figure 7 visualizes two typical cases (median and worst) for illustration.
+Structure segmentation from pathology enhanced images The anatomi-
+cal segmentation from pathology enhanced images, i.e., LGE CMR of MSCMRseg
+dataset, was a more challenging task compared to that of ACDC dataset. This
+is because MSCMRseg was a smaller dataset (e.g.: 25 vs. 70 training subjects),
+and the image quality and appearance pattern of LGE CMR could be worse and
+more complex.
+Table 4 provides the quantitative results, and Figure 8 visualizes two special
+examples (median and worst) for demonstration. ZScribbleSeg achieved promis-
+ing performance and better Dice and HD results than the other SOTA methods
+for scribble supervised segmentation. Notice that for this particular challenging
+task, the two general semi-supervised segmentation methods, i.e., CVIR and
+nnPU, could not work properly, which was confirmed by the two failed segmen-
+tation examples visualized in Figure 8.
+Finally, similar to the results in previous study (Section 3.4), ZScribbleSeg
+and FullSupUNet could achieve less noisy segmentation, affirmed by the remark-
+able better HD results in Table 4. Hence, we second to the conclusion that the
+proposed ZScribbleNet received greatly augmented supervision and global shape
+information via the proposed efficient scribble modeling and prior regularization.
+
+CCCCCCCZScribbleSeg
+25
+Irregular segmentation For segmentation of objects with heterogeneous shape
+features, it becomes particularly challenging to learn the accurate shape infor-
+mation for inference. We evaluated ZScribbleSeg on such challenging task of ir-
+regular segmentation using myocardial pathology segmentation (MyoPS), where
+we removed the shape regularization term Lshape due to the nature of pathologies
+lacking such property.
+Table 5 shows the performance in detail, and Figure 9 visualizes two typical
+cases, i.e., median cases by average Dice scores of edema and scar segmenta-
+tion, respectively. One can find that the advantages of the proposed methodolo-
+gies were demonstrated evidently in such challenging task, as the performance
+gains, either in terms of Dice or HD, were significant from CycleMix, ShapePU
+and finally to ZScribbleSeg compared to PCE, WSL4, GatedCRF, CVIR and
+nnPU (p < 0.001). In fact, the scribble-supervised segmentation of edema by
+the compared five methods were failed, and so were the segmentation of scar
+for WSL4 and GatedCRF. This is illustrated in the visualized examples in Fig-
+ure 9. Although WSL4 and GatedCRF worked well, with scribble supervision, in
+the above two regular structure segmentation tasks, they suffered severely from
+noisy labels due to their dependence of training on pseudo labels, which leads to
+the failure of model training. Furthermore, due to the similar texture between
+edema and surrounding tissues in all imaging modalities, it could be extremely
+difficult to segment such pathology relying solely on training images without ro-
+bust estimation and regularization of class mixture ratios. One can see from the
+result that this failed all the five compared methods in edema segmentation. By
+contrast, ShapePU and ZScribbleSeg succeeded in this task thanks to their own
+methods of estimating class prior π and applying spatial regularization, which
+is affirmed by the fact that they both achieved good HDs comparable to that
+of FullSupUNet for scar and edema segmentation. Notice that CycleMix did not
+illustrate such good performance in terms of HDs, but it achieved comparable
+good Dice scores thanks to the adoption of supervision augmentation.
+Segmentation from natural scenes We further validated the broad utility
+of ZScribbleSeg on the human pose segmentation task of natural scene images.
+We applied all the methods on the PPSS dataset, which consists of pedestrian
+images with occlusions, generated by different cameras with different resolutions.
+Table 6 presents the details, together with the summarized results from pre-
+vious three studies, i.e., ACDC, MSCMRseg and MyoPS. Similar to the three
+medical image segmentation tasks, the model of ZScribbleSeg generalized well to
+this 3-channel colored natural image segmentation task, with the performance
+comparable to FullSupUNet and Dice accuracy setting new state of the art for
+scribble supervised segmentation.
+Figure 10 visualizes three special cases, i.e., the best, median and the worst
+cases according to the average Dice by all compared methods. One can see from
+the figures that ZScribbleNet performed robustly and generated realistic seg-
+mentation with less noisy results, particularly compared with other scribble su-
+pervised methods and the fully supervised one (FullSupUNet).
+
+26
+K Zhang & X Zhuang
+Image
+Ground Truth
+PCE
+CVIR
+nnPU
+ShapePU
+CycleMix
+ZScribbleSeg
+FullSupUNet
+WSL4
+Best case
+Median case
+Worst case
+0.721
+0.736
+0.688
+0.699
+0.745
+0.708
+0.781
+0.690
+0.821
+0.795
+0.814
+0.791
+0.817
+0.832
+0.860
+0.862
+0.869
+0.795
+0.871
+0.868
+0.885
+0.898
+0.908
+0.914
+Dice (Avg)
+Scribble
+Dice (Avg)
+Dice (Avg)
+Fig. 10. Visualization of results on PPSS dataset. The selected subjects were the best,
+median and worst cases by the average Dice scores of all compared methods.
+Table 6. Dice results of the 10 methods on the four datasets. Note that sizes of training
+sets are given in the brackets.
+Methods
+ACDC
+MSCMRseg
+MyoPS
+PPSS
+(70)
+(25)
+(20)
+(2828)
+PCE
+.770±.126
+.385±.243
+.281±.271
+.805±.063
+WSL4 [29]
+.792±.166
+.848±.076
+-
+.762±.045
+GatedCRF [31] .804±.135
+.825±.032
+-
+-
+MAAG [42]
+.816
+-
+-
+.746
+CVIR [14]
+.800±.130
+.368±.095
+.293±.263
+.809±.054
+nnPU [20]
+.828±.123
+.437±.115
+.308±.282
+.794±.055
+CycleMix [52]
+.833±.098
+.771±.069
+.588±.191
+.835±.050
+ShapePU [53]
+.848±.100
+.833±.082
+.587±.205
+.823±.055
+ZScribbleSeg
+.862±.086 .870±.058 .636±.188 .838±.050
+FullSupUNet
+.854±.113
+.852±.076
+.633±.202
+.843±.071
+4
+Conclusion
+In this work, we have presented a new framework for scribble-supervised segmen-
+tation, i.e., ZScribbleSeg, to integrate the efficient scribbles and prior regulariza-
+tion with implementation of a deep neural network (ZScribbleNet). ZScribbleSeg
+exploits the principles of ”good scribble annotations”, and effectively augments
+the scribble supervision of ZScribbleNet, via mixup-occlusion operations and
+
+ZScribbleSeg
+27
+global consistency regularization. Then, we explored to capture the global in-
+formation by incorporating the prior information, particularly with proposals
+of spatial prior loss and shape prior loss. The spatial prior loss was based on
+the estimated spatial energy and label class mixture proportions π. The former
+provides a new means to identify the probability of unlabeled pixels belonging to
+each class without directly using model predictions; and the latter was developed
+based on a novel estimation method and was aimed to correct the problematic
+prediction via the regularization of spatial prior loss.
+To examine to performance of ZScribbleSeg, we investigated a variety of seg-
+mentation tasks, including regular structural segmentation of cardiac ventricles
+from anatomical imaging data (using ACDC dataset), regular structural segmen-
+tation of pathology enhanced imaging data (MSCMRseg), irregular object seg-
+mentation from multi-modality imaging (MyoPS), and human pose segmentation
+from natural scenario (PPSS). Compared to others approaches, ZScribbleSeg has
+shown great competence and achieved comparable performance to the fully su-
+pervised UNet method. Particularly, thanks to the augmented supervision and
+prior regularization, ZScribbleSeg performed well and demonstrated reliability
+and generalizability in the scenarios with small training set (MSCMRseg task)
+and irregular structure segmentation (MyoPS task), both of which failed several
+other state-of-the-art approaches.
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+

D9FRT4oBgHgl3EQfxziA/content/tmp_files/2301.13643v1.pdf.txt

@@ -0,0 +1,1909 @@
+arXiv:2301.13643v1  [math.CA]  31 Jan 2023
+Some Expansion Formulas for Brenke
+Polynomial Sets
+Hamza Chaggara, Abdelhamid Gahami and Neila Ben
+Romdhane
+Last Revised:
+February 1, 2023
+Abstract. In this paper, we derive some explicit expansion formulas
+associated to Brenke polynomials using operational rules based on their
+corresponding generating functions. The obtained coefficients are ex-
+pressed either in terms of finite double sums or finite sums or sometimes
+in closed hypergeometric terms. The derived results are applied to Gen-
+eralized Gould-Hopper polynomials and Generalized Hermite polynomi-
+als introduced by Szeg¨o and Chihara. Some well-known duplication and
+convolution formulas are deduced as particular cases.
+Mathematics Subject Classification (2010). 33C45, 41A10, 41A58.
+Keywords. Brenke polynomials, Connection coefficients, Generalized
+Gould-Hopper polynomials, Generalized Hermite polynomials, Generat-
+ing functions, Linearization coefficients.
+Contents
+1.
+Introduction
+2
+2.
+Operators Associated to Brenke PSs
+4
+2.1.
+Transfer Operator Associated to two Brenke Polynomials
+4
+2.2.
+XD-Expansion of the Operator θ
+5
+2.3.
+Examples
+5
+2.3.1.
+Hypergeometric Transformation
+6
+2.3.2.
+Particular Hypergeometric Transformation
+7
+2.3.3.
+Dunkl Operator on the Real Line
+8
+3.
+Connection and Linearization Problems
+9
+3.1.
+Connection Problem
+9
+3.1.1.
+Explicit Expression of the Connection Coefficients
+10
+3.1.2.
+Connection between two Db-Appell PSs
+11
+3.1.3.
+Addition and Convolution Type Formulas
+11
+
+2
+H. Chaggara, A. Gahami and N. Ben Romdhane
+3.1.4.
+Duplication Formula
+11
+3.2.
+Linearization Problems
+12
+3.2.1.
+Appell Polynomials
+13
+3.2.2.
+Explicit Expression of the LC
+13
+4.
+Application to Generalized Gould-Hopper Polynomial Set
+14
+4.1.
+Connection Problem
+14
+4.2.
+Linearization Formula
+16
+4.3.
+Generalized Hermite Polynomials
+17
+References
+18
+1. Introduction
+Let P be the vector space of polynomials with coefficients in C. A polynomial
+sequence in P is called polynomial set (PS for short) if deg Pn = n, for all n.
+The connection and linearization problems are defined as follows.
+Given two PSs {Pn}n≥0 and {Qn}n≥0, the so-called connection problem be-
+tween them asks to find the coefficients Cm(n), called connection coefficients
+CC, in the expression
+Qn(x) =
+n
+�
+m=0
+Cm(n)Pm(x).
+(1.1)
+The particular cases Qn(x) = xn and Qn(x) = Pn(ax), a ̸= 0, in (1.1) are
+known, respectively, as the inversion formula for {Pn}n≥0 and the duplication
+or multiplication formula associated with {Pn}n≥0.
+Given three PSs {Pn}n≥0, {Rn}n≥0 and {Sn}n≥0, then for
+Qi+j(x) =
+Ri(x)Sj(x) in (1.1) we are faced to the general linearization
+problem
+Ri(x)Sj(x) =
+i+j
+�
+k=0
+Lij(k)Pk(x).
+(1.2)
+The coefficients Lij(k) are called linearization coefficients LC.
+The particular case of this problem, Pn = Rn = Sn, is known as the standard
+linearization problem or Clebsch-Gordan-type problem.
+The computation and the positivity of the aforementioned coefficients
+play important roles in many situations of pure and applied mathemat-
+ics ranging from combinatorics and statistical mechanics to group theory
+[4, 21, 23]. Therefore, different methods have been developed in the litera-
+ture and several sufficient conditions for the sign properties to hold have
+been derived in [3, 31], using for this purpose specific properties of the in-
+volved polynomials such as orthogonality, generating functions, inversion for-
+mulas, hypergeometric expansion formulas, recurrence relations, algorithmic
+approaches, inverse relations,. . . (see e.g.[1, 2, 8, 13, 24, 32]). In particular, a
+general method, based on operational rules and generating functions, was
+
+Expansion Formulas for Brenke Polynomials
+3
+developed for polynomial sets with equivalent lowering operators and with
+Boas-Buck generating functions [6,12,14].
+In this paper, we deeply discuss both the connection and the lineariza-
+tion problems when the involved polynomials are of Brenke type. These poly-
+nomials are defined by their exponential generating functions as follows [9,17]
+A(t)B(xt) =
+∞
+�
+n=0
+Pn(x)
+n!
+tn,
+(1.3)
+where A and B are two formal power series satisfying:
+A(t) =
+∞
+�
+k=0
+aktk,
+B(t) =
+∞
+�
+k=0
+bktk,
+a0bk ̸= 0, ∀k ∈ N.
+(1.4)
+Brenke PSs are reduced to Appell ones when B = exp and they gener-
+ated many well-known polynomials in the literature, namely monomials,
+Hermite, Laguerre, Gould-Hopper, Generalized Hermite, Generalized Gould-
+Hopper, Appell-Dunkl, d-Hermite, d-Laguerre, Bernoulli, Euler, Al-Salam-
+Carlitz, Little q-Laguerre, q-Laguerre, discrete q-Hermite PSs,. . . .
+These polynomials appear in many areas of mathematics. In particular,
+in the framework of the standard orthogonality of polynomials, an exhaustive
+classification of all Brenke orthogonal polynomials was established by Chihara
+in [16]. Furthermore, Brenke polynomials play a central role in [25], where
+the authors determined all MRM-triples associated with Brenke-type gener-
+ating functions. Further, the positive approximation process discovered by
+Korovkin, a powerful criterion in order to decide whether a given sequence of
+positive linear operators on the space of continuous functions converges uni-
+formly in this space, plays a central role and arises naturally in many problems
+connected with functional analysis, harmonic analysis, measure theory, par-
+tial differential equations, and probability theory. The most useful examples
+of such operators are Sz´asz operators and many authors obtained a gener-
+alization of these operators using Brenke polynomials (see [33, 34] and the
+references therein).
+This paper is organized as follows. In Section 2, we define the transfer
+linear operator between two Brenke polynomials and which is illustrated by
+three interesting examples in particular the hypergeometric transformation
+and the Dunkl operator on the real line. Then in Section 3, we derive ex-
+pansion formulas associated to Brenke polynomials using operational rules
+and we give connection, linearization, inversion, duplication, and addition
+formulas corresponding to these polynomials. The obtained coefficients are
+expressed using generating functions involving the associated transfer lin-
+ear operators. Finally, in Section 4, we apply our obtained results to both
+Generalized Gould-Hopper PS (GGHPS) and Generalized Hermite PS (or
+Szeg¨o-Chihara PS) and we recover many known formulas as special cases.
+
+4
+H. Chaggara, A. Gahami and N. Ben Romdhane
+2. Operators Associated to Brenke PSs
+In this section, first, we introduce a transfer operator between two Brenke
+families, then we state its expression as an infinite series in the derivative
+operator D and the multiplication operator X known as XD-expansion [19].
+Finally, we give some examples.
+2.1. Transfer Operator Associated to two Brenke Polynomials
+Any Brenke PS {Pn}n≥0 generated by (1.3) is Db-Appell of transfer power
+series A, where A and b = (bn) are defined in (1.4). That is,
+DbPn+1 = (n + 1)Pn
+and
+A(Db)(bnxn) = Pn
+n! , n = 0, 1, 2, . . .,
+(2.1)
+where Db denotes the linear operator on P defined by [6]:
+Db(1) = 0, Db(xn) = bn−1
+bn
+xn−1, n = 1, 2, . . . .
+(2.2)
+The operator Db is known as the lowering operator for the PS {Pn}n≥0,
+however, A is the associated transfer series. (For more details, see [5]).
+Let {Pn}n≥0 and {Qn}n≥0 be two Brenke PSs generated respectively
+by:
+A1(t)B1(xt) =
+∞
+�
+n=0
+Pn(x)
+n!
+tn
+and
+A2(t)B2(xt) =
+∞
+�
+n=0
+Qn(x)
+n!
+tn,
+(2.3)
+where for i = 1, 2,
+Ai(t) =
+∞
+�
+k=0
+a(i)
+k tk,
+Bi(t) =
+∞
+�
+k=0
+b(i)
+k tk,
+a(i)
+0 b(i)
+k
+̸= 0, ∀ k ∈ N.
+(2.4)
+Then, the corresponding operators Db(1) and Db(2) are related by:
+Db(2)θ = θDb(1),
+(2.5)
+where θ is the bijective linear operator from P onto P (isomorphism of P)
+acting on monomials as follows:
+θ(xn) = b(2)
+n
+b(1)
+n
+xn
+and
+θ−1(xn) = b(1)
+n
+b(2)
+n
+xn.
+(2.6)
+The linear operator θ can be extended as a transfer operator taking any
+formal power series to another formal power series as follows
+θ(
+�
+n≥0
+anxn) =
+�
+n≥0
+anθ(xn),
+(2.7)
+and if φ(x) denotes a formal power series then one can easily check that,
+θ
+�
+φ(x)
+∞
+�
+k=0
+akxk�
+=
+∞
+�
+k=0
+akθ(φ(x)xk).
+(2.8)
+Hence, it is obvious that,
+θ(B1(x)) = B2(x).
+(2.9)
+
+Expansion Formulas for Brenke Polynomials
+5
+The operator θ will be called the transfer operator from B1 to B2 or transfer
+operator from {Pn}n≥0 to {Qn}n≥0.
+2.2. XD-Expansion of the Operator θ
+Now, recall that any operator L acting on formal power series has the follow-
+ing formal expansion, known as XD-expansion (see [19] and the references
+therein):
+L =
+∞
+�
+k=0
+Ak(X)Dk,
+(2.10)
+where D denotes the ordinary differentiation operator and {Ak(x)}k≥0 is a
+polynomial sequence such that:
+Lext =
+∞
+�
+k=0
+Ak(x)tkext.
+(2.11)
+We note that the infinite sum (2.10) is always well defined on P since when
+applied to any given polynomial, only a finite number of terms makes a
+nonzero contribution.
+The XD-expansion of the transfer operator θ is explicitly given by
+Proposition 2.1. The operator θ defined by (2.6) has the formal expansion:
+θ =
+∞
+�
+k=0
+φk
+k! XkDk,
+(2.12)
+where
+φk = (−1)k
+k
+�
+m=0
+(−k)m
+m!
+b(2)
+m
+b(1)
+m
+.
+Proof. By using (2.6) and (2.7) and then substituting L by θ in (2.11), we
+obtain
+θ(ext) =
+∞
+�
+k=0
+b(2)
+k
+b(1)
+k
+(xt)k
+k!
+=
+∞
+�
+k=0
+Ak(x)tkext.
+Therefore,
+∞
+�
+k=0
+Ak(x)tk = e−xt
+∞
+�
+k=0
+b(2)
+k
+b(1)
+k
+(xt)k
+k!
+=
+∞
+�
+k=0
+�
+k
+�
+m=0
+(−1)k (−k)m
+m!
+b(2)
+m
+b(1)
+m
+�
+(xt)k
+k!
+,
+which establishes the desired result.
+□
+2.3. Examples
+Here, we consider three interesting particular cases of the linear operator θ
+associated to two Brenke PSs and we essentially give integral representations
+for this operator.
+
+6
+H. Chaggara, A. Gahami and N. Ben Romdhane
+2.3.1. Hypergeometric Transformation. Recall first that rFs denotes
+the generalized hypergeometric function with r numerator parameters and s
+denominator parameters and defined as follows.
+rFs
+� (αr)
+(βs) ; x
+�
+=
+∞
+�
+k=0
+(α1)k(α2)k · · · (αr)k
+(β1)k(β2)k · · · (βs)k
+xk
+k! ,
+(2.13)
+where the contracted notation (αr) is used to abbreviate the array
+{α1, . . . , αr}, and (α)n denotes the Pochhammer symbol:
+(α)n = Γ(α + n)
+Γ(α)
+.
+(2.14)
+Consider two Brenke PSs {Pn}n≥0 and {Qn}n≥0 generated by (2.3) and (2.4)
+and such that the corresponding transfer linear operator θ takes the form:
+θ(xn) = b(2)
+n
+b(1)
+n
+xn = (γ1)n(γ2)n · · · (γp)n
+(δ1)n(δ2)n · · · (δp)n
+xn, γi ∈ C, δi ∈ C \ {−N}.
+(2.15)
+In this case, for the action of the operator θ on hypergeometric functions, we
+have the following result.
+Proposition 2.2. Let θ be defined by (2.15) with 0 < ℜ(γi) < ℜ(δi), then
+for r ≤ s + 1 and |x| < 1, we have
+θrFs
+�
+(αr)
+(βs) ; x
+�
+=
+p
+�
+i=1
+1
+β(γi, δi)
+�
+]0,1[p
+p
+�
+i=1
+uγi−1
+i
+(1 − ui)δi−γi−1
+× rFs
+�
+(αr)
+(βs) ; x
+p
+�
+i=1
+ui
+�
+du1 · · · dup,
+(2.16)
+where β designates the usual Euler’s Beta function,
+β(γ, δ) =
+� 1
+0
+tγ−1(1 − t)δ−1dt = Γ(γ)Γ(δ)
+Γ(γ + δ) , ℜ(γ), ℜ(δ) > 0.
+(2.17)
+Proof. From (2.7) and (2.15), we have
+θrFs
+�
+(αr)
+(βs) ; x
+�
+= p+rFp+s
+�
+(αr), (γp)
+(βs), (δp) ; x
+�
+.
+Thus, by using the Euler integral representation of generalized hypergeomet-
+ric functions, we obtain (see [27, p. 85]):
+p+rFp+s
+�
+(αr), (γp)
+(βs), (δp) ; x
+�
+=
+Γ(δp)
+Γ(γp)Γ(δp − γp)
+� 1
+0
+uδp−1
+p
+(1 − up)γp−δp−1
+× p+r−1Fp+s−1
+�
+(αr), (γp−1)
+(βs), (δp−1) ; xup
+�
+dup,
+and after (p − 1) similar applications of the Euler integral representation we
+get the desired result.
+□
+
+Expansion Formulas for Brenke Polynomials
+7
+When the operator θ is given by (2.15), the coefficient φk in Proposi-
+tion 2.1 is
+φk = (−1)k
+k
+�
+m=0
+(−k)m
+(γ1)m(γ2)m · · · (γp)m
+m!(δ1)m(δ2)m · · · (δp)m
+= (−1)kip+1Fp
+�
+−k, γ1, γ2, . . . , γp
+δ1, δ2, . . . , δp
+; 1
+�
+.
+Thus the corresponding XD expansion is
+θ =
+∞
+�
+k=0
+(−1)k
+k!
+p+1Fp
+�
+−k, γ1, γ2, . . . , γp
+δ1, δ2, . . . , δp
+; 1
+�
+XkDk.
+(2.18)
+2.3.2. Particular Hypergeometric Transformation. Here, we consider
+the special case θ(xn) = (γ)n
+(δ)n
+xn, δ ̸= 0, −1, −2, . . ..
+Proposition 2.3. For any analytic function f on ] − 1, 1[, f(x) =
+∞
+�
+n=0
+anxn,
+we have
+θ(f)(x) =
+1
+β(γ, δ − γ)
+� 1
+0
+tγ−1(1−t)δ−γ−1f(xt)dt, 0 < ℜ(γ) < ℜ(δ). (2.19)
+Moreover, the XD-expansion of θ is the following
+θ =
+∞
+�
+k=0
+(−1)k
+k!
+(δ − γ)k
+(γ)k
+XkDk.
+(2.20)
+Proof. By using (2.14) and (2.17), we obtain
+(γ)n
+(δ)n
+xn = Γ(γ + n)
+Γ(δ + n)
+Γ(δ)
+Γ(γ)xn =
+1
+β(γ, δ − γ)
+� 1
+0
+tγ−1(1 − t)δ−γ−1(xt)ndt.
+Thus, substituting the above equation in (2.7), we obtain (2.19) since the
+term-by-term integration is justified by the convergence of the series
+�
+n≥0
+� 1
+0
+��antγ−1(1 − t)δ−γ−1(xt)n�� dt.
+For (2.20), we use (2.18) and the Chu-Vandermonde reduction formula:
+2F1
+� −k, γ
+δ
+; 1
+�
+= (δ − γ)k
+(δ)k
+,
+δ ̸= 0, −1, −2, . . ..
+(2.21)
+Thus the proof is completed.
+□
+
+8
+H. Chaggara, A. Gahami and N. Ben Romdhane
+2.3.3. Dunkl Operator on the Real Line. The well-known Dunkl oper-
+ator, Dµ, associated with the parameter µ on the real line provides a useful
+tool in the study of special functions with root systems associated with finite
+reflection groups [20] and it is closely related to certain representations of
+degenerate affine Heke algebras [26]. This operator is defined by [20]:
+Dµ(f)(x) = Df(x) + µ
+x(f(x) − f(−x)),
+µ ∈ C,
+(2.22)
+where f is a real variable complex-valued function and D is the differentiation
+operator.
+The Dunkl operator acts on monomials as follows:
+Dµ(xn) =
+γµ(n)
+γµ(n − 1)xn−1, µ ̸= −1
+2, −3
+2, . . . ,
+(2.23)
+where
+γµ(2p + ǫ) = 22p+ǫp!(µ + 1
+2)p+ǫ,
+ǫ = 0, 1.
+(2.24)
+Hence, Dµ is a Db-operator type with bn =
+1
+γµ(n), and we have the following
+result.
+Proposition 2.4. Let µ1 and µ2 be two real numbers satisfying −1
+2 < µ1 <
+µ2, and θ given by
+θ(xn) = γµ1(n)
+γµ2(n)xn.
+(2.25)
+Then, for any analytic function, f on ] − 1, 1[, the following integral repre-
+sentation of θ holds true
+θ(f)(x) =
+1
+β(µ1 + 1
+2, µ2 − µ1)
+� 1
+−1
+f(xt)|t|2µ1(1 − t)µ2−µ1−1(1 + t)µ2−µ1 dt.
+(2.26)
+Proof. By using (2.14), (2.17) and (2.24) with µ replaced by µ1 and µ2, and
+for n = 2p + ǫ, ǫ = 0, 1, we obtain:
+γµ1(n)
+γµ2(n) = β(µ1 + 1
+2 + p + ǫ, µ2 − µ1)
+β(µ1 + 1
+2, µ2 − µ1)
+.
+(2.27)
+Now, with the beta integral representation (2.17), we get
+β(µ1 + 1
+2 + p + ǫ, µ2 − µ1) =
+� 1
+0
+tµ1+p+ǫ− 1
+2 (1 − t)µ2−µ1−1 dt,
+which, after the substitution u2 = t, and the distinction of the two cases
+ǫ = 0 and ǫ = 1, becomes
+β(µ1 + 1
+2 + p + ǫ, µ2 − µ1) =
+� 1
+−1
+un|u|2µ1(1 − µ)µ2−µ1−1(1 + u)µ2−µ1 du.
+
+Expansion Formulas for Brenke Polynomials
+9
+Consequently, this gives
+θ(xn) =
+1
+β(µ1 + 1
+2, µ2 − µ1)
+� 1
+−1
+(xt)n|t|2µ1(1 − t)µ2−µ1−1(1 + t)µ2−µ1 dt,
+(2.28)
+and a term-by-term integration achieves the proof.
+□
+The following two particular cases are worthy to note.
+• For f = expµ1, and according to (2.9), it is clear that
+θ(expµ1) = expµ2,
+where the generalized exponential function, expµ is defined by [28]
+expµ(x) =
+∞
+�
+n=0
+xn
+γµ(n),
+µ ̸= −1
+2, −3
+2, −5
+2, . . . .
+(2.29)
+So, for −1
+2 < µ1 < µ2, and by virtue of (2.26), the following integral
+representation of expµ2 holds true [28, Eq. (2.3.4)]:
+expµ2(x) =
+1
+β(µ1 + 1
+2, µ2 − µ1)×
+� 1
+−1
+expµ1(xt)|t|2µ1(1 − t)µ2−µ1−1(1 + t)µ2−µ1 dt.
+• For µ1 = 0 and µ2 = µ > 0, the transfer operator θ reduces to the well-
+known Dunkl intertwining operator Vµ in the one dimensional case and
+(2.26) is nothing else that its corresponding integral representation [20,
+Theorem 5.1]:
+Vµ(f)(x) =
+1
+β( 1
+2, µ)
+� 1
+−1
+f(xt)(1 − t)µ−1(1 + t)µ dt.
+(2.30)
+3. Connection and Linearization Problems
+In this section, we investigate connection and linearization formulas for
+Brenke PSs.
+3.1. Connection Problem
+Next, for two polynomial sequences of Brenke type, we state a generating
+function for the connection coefficients using the operator θ. This result ap-
+pears to be new. Some applications are given.
+Theorem 3.1. Let {Pn}n≥0 and {Qn}n≥0 be two polynomial sequences gen-
+erated by (2.3) and (2.4) and let θ be the corresponding transfer operator
+defined in (2.6). Then the CC in (1.1), (Cm(n))n≥m≥0, are generated by:
+A2(t)θ
+� tm
+A1(t)
+�
+=
+∞
+�
+n=m
+m!
+n! Cm(n)tn.
+(3.1)
+
+10
+H. Chaggara, A. Gahami and N. Ben Romdhane
+Proof. On one hand, substituting (1.1) in (2.3) and using sum manipulations,
+we get:
+A2(t)B2(xt) =
+∞
+�
+n=0
+Qn(x)tn
+n! =
+∞
+�
+n=0
+�
+n
+�
+m=0
+Cm(n)Pm(x)
+�
+tn
+n!
+=
+∞
+�
+m=0
+� ∞
+�
+n=m
+m!
+n! Cm(n)tn
+�
+Pm(x)
+m!
+.
+On the other hand, from (2.8), we have
+A2(t)B2(xt) = A2(t)θtB1(xt) = A2(t)θt
+�
+1
+A1(t)
+∞
+�
+m=0
+Pm(x)tm
+m!
+�
+=
+∞
+�
+m=0
+A2(t)θt
+� tm
+A1(t)
+� Pm(x)
+m!
+.
+Thus (3.1) follows and the proof is completed.
+□
+Some known results can be deduced from Theorem 3.1. Next, we quote
+the four important ones of them.
+3.1.1. Explicit Expression of the Connection Coefficients.
+Write
+1
+A1(t) =
+∞
+�
+n=0
+�a(1)
+n tn, then
+θt
+� tm
+A1(t)
+�
+=
+∞
+�
+n=0
+b(2)
+n+m
+b(1)
+n+m
+�a(1)
+n tn+m.
+By virtue of (3.1), we get:
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+� ∞
+�
+n=0
+a(2)
+n tn
+� � ∞
+�
+n=0
+b(2)
+n+m
+b(1)
+n+m
+�a(1)
+n tn+m
+�
+= tm
+∞
+�
+n=0
+� n
+�
+k=0
+a(2)
+k
+b(2)
+n+m−k
+b(1)
+n+m−k
+�a(1)
+n−k
+�
+tn
+=
+∞
+�
+n=m
+�n−m
+�
+k=0
+b(2)
+n−k
+b(1)
+n−k
+a(2)
+k �a(1)
+n−m−k
+�
+tn.
+Thus,
+Cm(n) = n!
+m!
+n−m
+�
+k=0
+b(2)
+n−k
+b(1)
+n−k
+a(2)
+k �a(1)
+n−m−k,
+m = 0, . . . , n.
+(3.2)
+In particular, we can deduce the explicit expansion and the inversion formula
+for any Brenke PS {Pn}n≥0 generated by (1.3):
+Pn(x)
+n!
+=
+n
+�
+m=0
+bman−mxm,
+and
+bnxn =
+n
+�
+m=0
+�an−m
+Pm(x)
+m!
+.
+(3.3)
+
+Expansion Formulas for Brenke Polynomials
+11
+3.1.2. Connection between two Db-Appell PSs. If B1 = B2, in (2.3),
+then by using (2.6), we obtain that the expression (3.1) takes the following
+simpler form [11].
+A2(t)
+A1(t) =
+∞
+�
+n=m
+m!
+n! Cm(n)tn−m.
+(3.4)
+3.1.3. Addition and Convolution Type Formulas. The Brenke PS {Pn}n≥0
+generated by (1.3) possesses the following generalized addition formula and
+convolution type relation:
+T b
+yPn(x) =
+n
+�
+m=0
+n!
+m!bn−myn−mPm(x),
+and
+A(Db)T b
+yPn(x) =
+n
+�
+m=0
+�n
+m
+�
+Pn−m(y)Pm(x),
+where T b
+y = B(yDb) designates the generalized translation operator satisfying
+T b
+y(B(xt) = B(yt)B(xt).
+In fact, for the addition formula, we remark that the PS, {T b
+yPn(x)}n≥0,
+is generated by:
+B(yt)A(t)B(xt) =
+∞
+�
+n=0
+T b
+yPn(x)
+n!
+tn,
+then we apply (3.4) with A2(t) = B(yt)A(t) and A1(t) = A(t), to obtain
+Cm(n) = n!
+m!bn−myn−m.
+For the convolution type relation, we apply the operator A(Db) to each
+member of the addition formula and we use (2.1). We have
+A(Db)T b
+yPn(x) =
+n
+�
+m=0
+n!
+m!(n − m)!A(Db)((n − m)!bn−myn−m)Pm(x)
+=
+n
+�
+m=0
+�n
+m
+�
+Pn−m(y)Pm(x).
+3.1.4. Duplication Formula. Brenke PS generated by (1.3) possesses the
+following duplication formula [11]
+Pn(ax) =
+n
+�
+m=0
+n!
+m!amβn−mPm(x),
+a ̸= 0,
+(3.5)
+where A(t)
+A(at) =
+∞
+�
+k=0
+βktk.
+In fact, the PS Qn(x) = Pn(ax) is generated by
+A(t)B(axt) =
+∞
+�
+n=0
+Qn(x)
+n!
+tn.
+
+12
+H. Chaggara, A. Gahami and N. Ben Romdhane
+Thus, by using (2.6) and (2.7), we have θ(f)(x) = f(ax), where f is any
+formal power series.
+Now, from (3.1), with A1(t) = A2(t) = A(t), it follows immediately that
+(at)m A(t)
+A(at) =
+∞
+�
+n=m
+m!
+n! Cm(n)tn.
+3.2. Linearization Problems
+In the following result, we provide a generating function for the LC involving
+three Brenke polynomials.
+Theorem 3.2. Let {Pn}n≥0, {Rn}n≥0 and {Sn}n≥0 be three Brenke PS with
+exponential generating functions:
+A1(t)B1(xt), A2(t)B2(xt) and A3(t)B3(xt),
+(3.6)
+where Ai(t) =
+∞
+�
+k=0
+a(i)
+k tk, Bi(t) =
+∞
+�
+k=0
+b(i)
+k tk, a(i)
+0 b(i)
+k
+̸= 0, ∀k ∈ N, i = 1, 2, 3.
+Then the LC, {Lij(k)}i,j≥0, k ∈ N, defined in (1.2) are generated by:
+A2(s)A3(t)
+k!
+θ(2)
+s θ(3)
+t
+(θ(1)
+s+t)−1
+� (s + t)k
+A1(s + t)
+�
+=
+�
+i,j≥0
+Lij(k)
+i!j! sitj
+(3.7)
+where θ(i)(tn) = n!b(i)
+n tn,
+i = 1, 2, 3.
+We note that θ(i), i = 1, 2, 3, are the transfer operators from {Pn}n≥0,
+{Rn}n≥0 and {Sn}n≥0, to the monomials, respectively.
+Proof. On one hand, according to (1.2) and with sum manipulation, we ob-
+tain:
+�
+i,j≥0
+Ri(x)Sj(x)si
+i!
+tj
+j! =
+�
+i,j≥0
+�i+j
+�
+k=0
+Lij(k)Pk(x)
+�
+si
+i!
+tj
+j!
+=
+∞
+�
+k=0
+
+k!
+�
+i,j≥0
+Lij(k)
+i!j! sitj
+
+ Pk(x)
+k!
+.
+(3.8)
+On the other hand, by using (2.6), we can easily verify that
+θ(2)
+s θ(3)
+t (θ(1)
+s+t)−1B1((s + t)x) =
+∞
+�
+k=0
+� k
+�
+l=0
+b(2)
+l
+b(3)
+k−lsltk−l
+�
+xk,
+then
+B2(xs)B3(xt) = θ(2)
+s θ(3)
+t (θ(1)
+s+t)−1B1((s + t)x).
+Using the generating function of {Pn}n≥0, we obtain
+B2(xs)B3(xt) =
+∞
+�
+k=0
+�
+θ(2)
+s θ(3)
+t
+(θ(1)
+s+t)−1 (s + t)k
+A1(s + t)
+� Pk(x)
+k!
+.
+
+Expansion Formulas for Brenke Polynomials
+13
+Thus
+�
+i,j≥0
+Ri(x)Sj(x)si
+i!
+tj
+j! =
+∞
+�
+k=0
+�
+A2(s)A3(t)θ(2)
+s θ(3)
+t (θ(1)
+s+t)−1 (s + t)k
+A1(s + t)
+� Pk(x)
+k!
+.
+Equating the coefficients of Pk(x) in the above equation and (3.8), we obtain
+(3.7) which finishes the proof.
+□
+Next, as applications, we recover the generating function for the LC of
+three Appell polynomials and the explicit expression of the LC associated to
+three Brenke PS.
+3.2.1. Appell Polynomials. Let {Pn}n≥0, {Rn}n≥0, and {Sn}n≥0, be three
+Appell-PS. Then we have B1 = B2 = B3 = exp, and by applying Theo-
+rem 3.2, we obtain that the LC in (1.2) are generated by
+A2(s)A3(t)
+A1(s + t)
+(s + t)k
+k!
+=
+∞
+�
+i,j=0
+Lij(k)
+i!j! sitj,
+(3.9)
+which agrees with Carlitz Formula [10, Eq.(1.9)].
+Moreover, for Pn = Rn = Sn = Hn, where Hn are Hermite polynomials
+generated by
+e−t2e2xt =
+∞
+�
+n=0
+Hn(x)tn
+n!,
+(3.10)
+we have A1(t) = A2(t) = A3(t) = A(t) = e−t2, and then
+A(s)A(t)
+A(s + t)
+(s + t)k
+k!
+= 1
+k!e2st(s + t)k.
+Thus, using (3.9) we deduce the standard linearization formula for Hermite
+PSs
+Hi(x)Hj(x) =
+min(i,j)
+�
+k=0
+�i
+k
+��j
+k
+�
+2kk!Hi+j−2k(x).
+(3.11)
+This formula is known as Feldheim formula [3].
+3.2.2. Explicit Expression of the LC. For three Brenke PS satisfying
+the hypothesises of Theorem 3.2, the LC in (1.2) are given by:
+Lij(k) = i!j!
+k!
+i
+�
+n=0
+j
+�
+m=0
+b(2)
+n b(3)
+m
+b(1)
+n+m
+a(2)
+i−na(3)
+j−m�a(1)
+n+m−k,
+k = 0, 1, . . . , i + j, (3.12)
+where 1/A1(t) =
+∞
+�
+n=0
+�a(1)
+n tn, and
+�a(1)
+−n = 0, n = 1, 2, . . . .
+Indeed, we have (s + t)k
+A1(s + t) =
+∞
+�
+n=k
+�a(1)
+n−k(s + t)n, then by using (2.6), we get
+θ(2)
+s θ(3)
+t (θ(1)
+s+t)−1
+� (s + t)k
+A1(s + t)
+�
+=
+∞
+�
+n=k
+�a(1)
+n−k
+n
+�
+m=0
+b(2)
+n−mb(3)
+m
+b(1)
+n
+tmsn−m.
+
+14
+H. Chaggara, A. Gahami and N. Ben Romdhane
+Thus, with sum manipulations and (3.7), one can easily verify that
+�
+i,j≥0
+Lij(k)
+i!j! sitj = 1
+k!
+∞
+�
+n,m=0
+� ∞
+�
+i=n
+a(2)
+i−nsi
+� 
+
+∞
+�
+j=m
+a(3)
+j−mtj
+
+ b(2)
+n b(3)
+m
+b(1)
+n+m
+�a(1)
+n+m−k
+= 1
+k!
+�
+i,j≥0
+�
+i
+�
+n=0
+j
+�
+m=0
+b(2)
+n b(3)
+m
+b(1)
+n+m
+a(2)
+i−na(3)
+j−m�a(1)
+n+m−k
+�
+sitj,
+which leads to (3.12).
+We note that this result was first obtained in [11, Corollary 3.3] by using
+a method based on the inversion formula.
+4. Application to Generalized Gould-Hopper
+Polynomial Set
+The (d + 1)-fold symmetric generalized Gould-Hopper polynomials,
+{Q(d+1)
+n
+(·, a, µ)}n≥0, are generated by [7]:
+eatd+1 expµ(xt) =
+∞
+�
+n=0
+Q(d+1)
+n
+(x, a, µ)
+n!
+tn, a ∈ C, µ ̸= −1
+2, −3
+2, −5
+2, . . . , (4.1)
+where a PS {Pn}n≥0 is said to be (d + 1)-fold symmetric, d = 1, 2, . . . , if
+Pn
+�
+e
+2iπ
+d+1 x
+�
+= e
+2inπ
+d+1 Pn(x).
+These polynomials constitute a unification of many known families such as:
+• Classical Hermite PS, Hn(x) = Q(2)
+n (2x, −1, 0).
+• Gould-Hopper PS, gm
+n (x, h) = Q(m)
+n
+(x, h, 0), (same notations as in [22]).
+• Generalized Hermite polynomials [30]:
+Hµ
+n(x) = Q(2)
+n (2x, −1, µ).
+(4.2)
+The GGHPS are of Brenke type with transfer power series A(t) = exp(atd+1).
+They are the only (d + 1)-fold symmetric Dunkl-Appell d-orthogonal PS [7].
+Next, we solve the connection and linearization problems associated to
+GGHPS and we treat the particular case of generalized Hermite polynomials.
+4.1. Connection Problem
+Here, we state the connection formulas for two GGHPS when one or two
+of the parameters are different and we give an integral representation of
+these coefficients. Moreover, the inversion formula, addition and convolution
+relations, and duplication formula are given.
+Theorem 4.1. The connection coefficients, Cn−i(d+1)(n), 0 ≤ i ≤ [
+n
+d + 1],
+between two GGHPS, {Q(d+1)
+n
+(·, a, µ1)}n≥0 and {Q(d+1)
+n
+(·, b, µ2)}n≥0 are given
+
+Expansion Formulas for Brenke Polynomials
+15
+by
+Cn−i(d+1)(n) =
+n!
+(n − i(d + 1))!
+i
+�
+k=0
+γµ1(n − k(d + 1))
+γµ2(n − k(d + 1))
+(−a)i−k
+(i − k)!
+bk
+k! .
+(4.3)
+Proof. By means of (2.6), we have
+θ(tme−atd+1) =
+∞
+�
+n=0
+(−a)n
+n!
+γµ1(n(d + 1) + m)
+γµ2(n(d + 1) + m)tn(d+1)+m.
+Thus, by using (3.1), (4.1) and sum manipulation, we obtain
+∞
+�
+n=m
+m!
+n! Cm(n)tn = ebtd+1θ(tme−atd+1)
+=
+∞
+�
+i=0
+1
+i!
+i
+�
+k=0
+�i
+k
+�γµ1(k(d + 1) + m)
+γµ2(k(d + 1) + m)bi−k(−a)k ti(d+1)+m.
+Therefore, for n = i(d + 1) + m, the desired result holds.
+□
+We note that for the particular case µ1 = µ2, (4.3) is reduced to
+Cn−i(d+1)(n) =
+n!(b − a)i
+i!(n − i(d + 1))!,
+0 ≤ i ≤
+�
+n
+d + 1
+�
+.
+For the connection coefficients obtained in Theorem 4.3, we have the following
+result.
+Proposition 4.2. For µ2 > µ1 > −1
+2, the connection coefficient given by
+(4.3) has the following integral representation,
+Cn−i(d+1)(n) = n!β−1(µ1 + 1
+2, µ2 − µ1)
+i!(n − i(d + 1))!
+×
+� 1
+−1
+tn−i(d+1)|t|2µ1(b − atd+1)i (1 − t2)µ2−µ1
+1 − t
+dt.
+Proof. Using Proposition 2.4 with f(x) = xn−k(d+1) and x = 1, we obtain
+γµ1(n − k(d + 1))
+γµ2(n − k(d + 1)) =
+1
+β(µ1 + 1
+2, µ2 − µ1)
+� 1
+−1
+tn−k(d+1)|t|2µ1 (1 − t2)µ2−µ1
+1 − t
+dt.
+Substituting the above equation in (4.3), we get:
+Cn−i(d+1)(n) =
+n!
+i!(n − i(d + 1))!
+1
+β(µ1 + 1
+2, µ2 − µ1)×
+� 1
+−1
+tn|t|2µ1 (1 − t2)µ2−µ1
+1 − t
+�
+i
+�
+k=0
+�i
+k
+�
+(−a)i−k(
+b
+td+1 )k
+�
+dt,
+from which the desired result follows.
+□
+Next, we give some specific expansion relations associated to GGHPS.
+
+16
+H. Chaggara, A. Gahami and N. Ben Romdhane
+• Explicit and inversion formulas: The following explicit expression and
+inversion formula of {Q(d+1)
+n
+(·, a, µ}n≥0 can be easily derived from (3.3):
+Q(d+1)
+n
+(x, a, µ) = n!
+[
+n
+d+1 ]
+�
+k=0
+ak
+k!γµ(n − (d + 1)k) xn−(d+1)k,
+(4.4)
+and
+xn
+γµ(n) =
+[
+n
+d+1 ]
+�
+k=0
+(−a)k
+k!(n − (d + 1)k)!Q(d+1)
+n−(d+1)k(x, a, µ).
+(4.5)
+• Addition and convolution relations:
+T µ
+y Q(d+1)
+n
+(x, a, µ) =
+n
+�
+k=0
+n!yn−k
+k!γµ(n − k)Q(d+1)
+k
+(x, a, µ),
+(4.6)
+2
+n
+d+1 T µ
+y Q(d+1)
+n
+�
+2
+−1
+d+1 x, a, µ
+�
+=
+n
+�
+k=0
+�n
+k
+�
+Q(d+1)
+k
+(y, a, µ) Q(d+1)
+n−k (x, a, µ), (4.7)
+where T µ
+y = expµ(yDµ).
+For µ = 0, this equation is reduced to the well-known Gould-
+Hopper convolution type relation [22] and for m = 2, h = −1, we recover
+the Runge formula for Hermite polynomials [29]
+• Duplication formula:
+Q(d+1)
+n
+(αx, a, µ) = n!
+[
+n
+d+1 ]
+�
+k=0
+αn−k(d+1)(1 − αd+1)kak
+(n − k(d + 1))!k!
+Q(d+1)
+n−k(d+1)(x, a, µ), α ̸= 0.
+4.2. Linearization Formula
+Taking into account the (d + 1)-fold symmetry property of the GGHPS, any
+LC Lij(k), in (1.2) vanishes when k ̸= i + j − r(d + 1). Thus, according to
+(3.12), the corresponding LC is given by:
+Lij(i + j − r(d + 1)) =
+i!j!
+(i + j − r(d + 1))!
+[
+i
+d+1 ]
+�
+n=0
+[
+j
+d+1 ]
+�
+m=0
+an
+1am
+2 (−a3)r−m−n
+n!m!(r − m − n)! ×
+γµ3(i + j − (m + n)(d + 1))
+γµ1(i − n(d + 1))γµ2(j − r(d + 1)), 0 ≤ r ≤
+� i + j
+d + 1
+�
+.
+
+Expansion Formulas for Brenke Polynomials
+17
+We remark that there is no difficulty in proving the corresponding formula
+for the linearization of any arbitrary number of GGHPSs. We have:
+N
+�
+s=1
+Q(d+1)
+is
+(x, as, µs) =
+[ i1+···+iN
+d+1
+]
+�
+r=0
+i1! · · · iN!
+(i1 + · · · + iN − r(d + 1))!×
+[
+i1
+d+1 ]
+�
+s1=0
+· · ·
+[ iN
+d+1 ]
+�
+sN =0
+as1
+1 · · · asN
+N (−aN+1)r−s1−···−sN
+s1! · · · sN!(r − s1 − · · · − sN)! ×
+γµN+1(i1 + · · · + iN − (d + 1)(s1 + · · · + sN))
+γµ1(i1 − (d + 1)s1) · · · γµN(iN − (d + 1)sN) ×
+Q(d+1)
+i1+···+iN −r(d+1)(x, aN+1, µN+1).
+4.3. Generalized Hermite Polynomials
+The generalized Hermite polynomials, {Hµ
+n}n≥0, are introduced by Szeg¨o [30],
+then investigated by Chihara in his PhD Thesis [15] and further studied by
+many other authors [11,28]. They are generated by:
+e−td+1 expµ(2xt) =
+∞
+�
+n=0
+Hµ
+n(x)
+n!
+tn, µ ̸= −1
+2, −3
+2, −5
+2, . . . .
+(4.8)
+Proposition 4.3. The following connection relation holds:
+�Hµ2
+n (x) =
+[n/2]
+�
+k=0
+(−1)k 4k
+k!
+(µ2 − µ1)k �Hµ1
+n−2k(x), µ2 > µ1 > −1
+2,
+(4.9)
+where { �Hµi
+n }n, i = 1, 2 are the normalized generalized Hermite PS given by
+�Hµi
+n (x) = γµi(n)
+n![ n
+2 ]! Hµi
+n (x).
+Proof. From what has already been stated, the connection coefficients from
+{Hµ2
+n }n to {Hµ1
+n }n are generated by
+e−t2θ(tmet2) =
+∞
+�
+n=m
+m!
+n! Cm(n)tn,
+where θ is the operator defined in (2.25).
+Making use of the θ-integral representation (2.26), intercalate 0 in the interval
+of integration, we get:
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+tme−t2
+β(µ1 + 1
+2, µ2 − µ1)×
+� 1
+0
+et2s2
+sm+2µ1
+(1 − s2)µ1−µ2
+�
+1
+1 − s + (−1)m
+1 + s
+�
+ds.
+
+18
+H. Chaggara, A. Gahami and N. Ben Romdhane
+It follows, for m even and after substituting u = s2, that
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+tme−t2
+β(µ1 + 1
+2, µ2 − µ1)
+� 1
+0
+eut2u
+m−1
+2
++µ1(1 − u)µ2−µ1−1du
+=
+∞
+�
+n=0
+(−1)n
+n!
+β(µ1 + m+1
+2 , µ2 − µ1 + n)
+β(µ1 + 1
+2, µ2 − µ1)
+tm+2n,
+where the term by term integration is justified by the same argument as in
+the proof of Proposition 2.3.
+On the other hand, we have
+β(µ1 + 1
+2 + k, µ2 − µ1 + n)
+β(µ1 + 1
+2, µ2 − µ1)
+= Γ(µ1 + 1
+2 + k)Γ(µ2 − µ1 + n)Γ(µ2 + 1
+2)
+Γ(µ2 + n + k + 1
+2)Γ(µ1 + 1
+2)Γ(µ2 − µ1)
+= γµ1(2k)
+22kk!
+22(k+n)(k + n)!
+γµ2(2(k + n)) (µ2 − µ1)n
+=
+γµ1(m)
+γµ2(m + 2n)
+4n([m/2] + n)!
+[m/2]!
+(µ2 − µ1)n.
+Thus, by virtue of (2.17) and (2.27), we obtain
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+∞
+�
+n=0
+(−1)n
+n!
+γµ1(m)4n([ m
+2 ] + n)!
+γµ2(m + 2n)[ m
+2 ]! (µ2 − µ1)ntm+2n.
+For m odd, similar computations lead to
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+∞
+�
+n=0
+(−1)n
+n!
+γµ1(m)
+γµ2(m + 2n)
+4n([ m
+2 ] + n)!
+[ m
+2 ]!
+(µ2 − µ1)ntm+2n.
+Therefore, for m = 0, 1, 2, 3, . . ., we have:
+∞
+�
+n=m
+m!
+n! Cm(n)tn =
+∞
+�
+n=0
+(−1)n
+n!
+γµ1(m)
+γµ2(m + 2n)
+4n([ m
+2 ] + n)!
+[ m
+2 ]!
+(µ2 − µ1)ntm+2n,
+Thus, for k = 0, 1, 2, . . ., [n
+2 ], we get
+Cn−2k(n) = (−1)k
+k!
+n!
+(n − 2k)!
+4k[ n
+2 ]!
+[ n
+2 − k]!
+γµ1(n − 2k)
+γµ2(n)
+(µ2 − µ1)k.
+□
+We note that the connection coefficients in (4.9) alternate in sign and
+that this relation was already derived in [14], where the authors used a linear
+computer algebra approach based on the Zeilberger’s algorithm.
+References
+[1] Abd-Elhameed, W., Badah, B.M.: New approaches to the general linearization
+problem of Jacobi polynomials based on moments and connection formulas.
+Mathematics 9, 1–28 (2021)
+
+Expansion Formulas for Brenke Polynomials
+19
+[2] Area,, I., Godoy, E., Rodal, J., Ronveaux, A., Zarzo A.: Bivariate Krawtchouk
+polynomials: Inversion and connection problems with the NAVIMA algorithm.
+J. Comput. Appl. Math. 284, 50–57 (2015)
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+sylvania (1975)
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+Transforms Spec. Funct. 18, 581–597 (2007)
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+sets. Med. J. Math. 13, 2783–2793 (2016)
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+(1963)
+[11] Chaggara, H.: Operational rules and a generalized Hermite polynomials. J.
+Math. Anal. Appl. 332, 11–21 (2007)
+[12] Chaggara, H.: Quasi monomialty and linearization coefficients for Sheffer poly-
+nomial sets. Difference Equations, Special Functions, And Orthogonal Polyno-
+mials pp. 90–99 (2007)
+[13] Chaggara, H., Mabrouk, M.: Linearization coefficients for some basic hyperge-
+ometric polynomials. J. Mathematics Volume 2022, 12 pages
+[14] Chaggara, H., Koepf, W.: On linearization and connection coefficients for gen-
+eralized Hermite polynomials. J. Math. Anal. Appl. 236, 65–73 (2011)
+[15] Chihara, T.: Generalized Hermite polynomials. Ph.D. thesis, Purdue (1955)
+[16] Chihara, T.: Orthogonal polynomials with Brenke type generating functions.
+Duke Math. J. 35, 505–517 (1968)
+[17] Chihara, T.: An Introduction to Orthogonal Polynomials. Gordon and Breach,
+New York, London, Paris (1978)
+[18] Dehesa, J., Martinez-Finkelshtein, A., S´anchez-Ruiz, J.: Quantum information
+entropies and orthogonal polynomials. J. Comput. Appl. Math. 133, 23–46
+(2001)
+[19] Di Bucchianico, A., Loeb, D.E.: Operator expansion in the derivative and mul-
+tiplication by x. Integral Transforms Spec. Funct. 4, 49–68 (1996)
+[20] Dunkl, C.: Integral kernels with reflection group invariance. Canad. J. Math.
+43, 1213–1227 (1991)
+[21] Gasper, G.: Linearization of the product of Jacobi polynomials. Canad. J.
+Math. 22, 171–175 (1970)
+[22] Gould, H., Hopper, A.T.: Operational formulas connected with two generaliza-
+tions of Hermite polynomials. Duke Math. J. 29, 51–63 (1962)
+
+20
+H. Chaggara, A. Gahami and N. Ben Romdhane
+[23] Koornwinder, T.: Compact quantum groups and q-special functions 311, 46–
+128 (1994)
+[24] Maroni, P., Da Rocha, Z.: Connection coefficients for orthogonal polynomials:
+symbolic computations, verification, and demonstrations in the Mathematica
+language. Numer. Algor. 63, 507–520 (2013)
+[25] Asai, N., Kubo, I., Kuo, H.H.: The Brenke type generating functions and ex-
+plicit forms of MRM-triples by means of q-hypergeometric series. Inf. Dimens.
+Anal. Quantum Probab. Related Topics, 16 27 pages (2013).
+[26] Opdam, E.M.: Dunkl operators, Bessel functions and the discriminant of a
+finite Coxeter group. Compos. Math., 85, 333–373 (1993).
+[27] Rainville, E.: Special Functions. The Macmillan Company, New York (1960)
+[28] Rosenblum, M.: Generalized Hermite polynomials and the Bose-like oscillator
+calculus. Oper. Theory Adv. Appl. 73, 369–396 (1994)
+[29] Runge, C.: ¨Uber eine besondere art von intergralgleichungen. Math. Ann. 75,
+130–132 (1914)
+[30] Szeg¨o, G. : Orthogonal polynomials, 4rd edn. Amer. Math. Soc. Colloq. Vol.
+23, Amer. Math. Soc, New York (1975)
+[31] Szwarc, R.: Convolution structures associated with orthogonal polynomials. J.
+Math. Anal. Appl. 170, 158–170 (1992)
+[32] Tcheutia,, D., Foupouagnigni, M., Koepf, W., Sadjang, N.N.: Coefficients of
+multiplication formulas for classical orthogonal polynomials. Ramanujan J. pp.
+1–35 (2015)
+[33] Varma, S., Sezgin, S., ´I¸c¨oz, G.: Generalization of Szasz operators involving
+Brenke type polynomials. Comput. Math. Appl. 64, 121–127 (2012)
+[34] Wani, S., Mursaleen, M., Nisar, K.S.: Certain approximation properties of
+Brenke polynomials using Jakimovski-Leviatan operators. J. Inequal. Appl.
+64, 1–16 (2021)
+Hamza Chaggara
+Mathematics Department, College of Science, King Khalid University, Abha, King-
+dom of Saudi Arabia/D´epartement de Math´ematiques, ´Ecole Sup´erieure des Sci-
+ences et de la Technologie, Sousse University, Tunisia.
+e-mail: hshaggara@kku.edu.sa / hamza.chaggara@ipeim.rnu.tn
+Abdelhamid Gahami
+D´epartement de Math´ematiques, Institut Pr´eparatoire aux ´Etudes d’Ing´enieur, Sfax
+University, Tunisia.
+e-mail: aelgahami@yahoo.fr
+Neila Ben Romdhane
+D´epartement de Math´ematiques, ´Ecole Sup´erieure des Sciences et de la Technologie,
+Sousse University, Tunisia.
+e-mail: neila.benromdhane@ipeim.rnu.tn
+

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+Organised Firestorm as strategy for business
+cyber-attacks
+Andrea Russo
+Department of Physics and Astronomy, University of Catania, Italy
+Email: andrea.russo@phd.unict.it
+Abstract—Having a good reputation is paramount for most or-
+ganisations and companies. In fact, having an optimal corporate
+image allows them to have better transaction relationships with
+various customers and partners. However, such reputation is hard
+to build and easy to destroy for all kind of business commercial
+activities (B2C, B2B, B2B2C, B2G). A misunderstanding during
+the communication process to the customers, or just a bad
+communication strategy, can lead to a disaster for the entire
+company. This is emphasised by the reaction of millions of
+people on social networks, which can be very detrimental for
+the corporate image if they react negatively to a certain event.
+This is called a firestorm.
+In this paper, I propose a well-organised strategy for firestorm
+attacks on organisations, also showing how an adversary can
+leverage them to obtain private information on the attacked
+firm. Standard business security procedures are not designed to
+operate against multi-domain attacks; therefore, I will show how
+it is possible to bypass the classic and advised security procedures
+by operating different kinds of attack. I also propose a different
+firestorm attack, targeting a specific business company network
+in an efficient way. Finally, I present defensive procedures to
+reduce the negative effect of firestorms on a company.
+Index
+Terms—Firestorm,
+Cyber-attack,
+Business
+Defence,
+Socio-dynamics, Stress Test, Network Science, Cyberpunk 2077.
+I. INTRODUCTION
+Before the advent of social medias, brand crises were largely
+caused by journalists’ contributions. Nowadays, a firestorm is
+a cluster of consumers’ digital word of mouth that highlights
+some communication error, or some terrible mistake made
+by a company [15]. The Cambridge dictionary1 defines the
+firestorm as “a sudden, and sometimes violent reaction” and
+the shitstorm as “a wildly chaotic and unmanageable situation,
+controversy, or sequence of events”. In this paper, I will use
+both these terms interchangeably.
+During the last years, many firestorms took place on the
+Internet [19], [27], [31], mainly due to the increase of the
+number of users on social networks. In some cases, firestorms
+have been formally studied to better understand this phe-
+nomenon [15], [28], [31]. In 2007, several researchers debated
+over firestorms, and one of the main outcomes is that “a
+natural science model of the research process is suitable for
+studying the social world but a central issue remaining of
+whether the social world can, and should be, studied according
+to the same principles, procedures, and philosophy as the
+natural sciences”
+[1]. This is relevant because today I are
+actually able to study and evaluate social dynamics by using
+1https://dictionary.cambridge.org
+the massive amount of data coming from the digital world,
+with particular emphasis on social networks [32].
+Firestorms are not made of a single event with a standard
+behaviour, instead they are caused by non-linear dynamics
+leading to complex behaviours. Due to this, companies must
+have appropriate procedures to respond to various crisis situa-
+tions. Lehtonen’s theory [23] shows that a firestorm develops
+in five stages: (1) latent stage, where weak signals of the
+upcoming crisis are received; (2) triggering event, where the
+subject becomes the target of news and social media attention;
+(3) the subject is in the top-news and the media attention
+spikes; (4) the media attention calms down to the level of
+general philosophical and ethical discussion; and (5) there
+are only minor media hits and attention is guided to other
+issues [28].
+As firestorms begin when there is a service failure, a
+social failure or when a company fails to communicate prop-
+erly [15], this kind of errors can be reduced by following
+appropriate procedures. However, most of the existing quality
+and security procedures, such as the ones suggested by ISO
+9001:2015 [17] and ISO/IEC 27002:2022 [18] are not ade-
+quate for a multi-domain cyber and social attack. Because,
+regard to the 27002:2022, social attacks are outside the scope,
+while, 9001:2015, even if it focuses on better business process
+quality, thus, less firestorm risk from the public, it does not
+mitigate the firestorm from an attacker.
+Hence, in this paper I theorise that it is possible for an
+attacker to intentionally cause a firestorm attack to undermine
+the reputation of a company, with the side-effect of advan-
+taging the competitors. I argue that self-organised Firestorm
+attacks require a high number of bots that are already active
+on social medias: in this case, bots start the firestorm on the
+target company, spreading fake news (or magnifying a certain
+event, e.g., a mistake made by the company in the past) that
+will cause a high volume of real people to react negatively
+and continue the social attack, unknowingly on behalf of the
+adversary.
+Additionally, I argue that Open Source Intelligence (OS-
+INT) could allow an adversary to identify weak spots in the
+organization, namely people who most likely cannot react
+properly or defend themselves from the firestorm, hence not
+being able to timely mitigate its impact. Many workers have a
+LinkedIn, Facebook, or Twitter account: moving the firestorm
+on the social media accounts of people who work for the
+target company can lead to an extremely stressful situation for
+arXiv:2301.01518v1  [cs.CY]  4 Jan 2023
+
+workers. This could be even worse for people who do not often
+deal with public relations, and could cause confusion, panic
+and distress. In fact, when a firestorm arises, even people who
+work on communication processes and managers can panic,
+and the fear of losing customers and partners can be very
+detrimental for any company.
+When people working in the target firm are in this altered
+status, I argue it is possible to elaborate a social engineering
+strategy to capture protected information: in this case, not only
+firestorms serve the purpose to undermine the corporate image,
+but they are also used as a diversion for a social engineering
+attack. In fact, while most important organisations adhere
+to best-practices listed in security standards like ISO/IEC
+27002:2022 [18], during a social attack like firestorms, some
+best-practices and procedures may be distorted or bypassed,
+both intentionally or by mistake, due to the pressure applied
+to people who are in charge of complying to such procedures
+[14].
+Contributions. The paper makes these contributions:
+1) I explain how to make an automated and organized
+firestorm attack, with only a few manual operations such
+as the choice of a topic and of a hashtag;
+2) I introduce a taxonomy of possible actions that the
+attacker could perform while doing the firestorm;
+3) I illustrate how the author of a firestorm can evade
+detection for their attack by targeting single workers
+instead of the company profiles, while increasing the
+damage done to the firm.
+4) I show possible long and short term procedures that
+a company can implement to mitigate the effect of
+firestorms attacks.
+II. CYBER-ATTACK PLANING PRELUDE
+In this section, I illustrate a novel strategy to artificially
+cause a firestorm, leveraging a botnet to start agitating real
+people against a target company. Due to the large number of
+posts that bots can create within seconds, they can be used
+to amplify any idea on social networks, influencing political
+affairs [3] and business company value [33]. For example,
+due to a cyber-attack on a Twitter newspaper profile, such
+newspaper shared a fake news about President Obama being
+injured by a bomb in the White House, causing a flash-crash
+in Wall Street and the stop all of economic transactions for
+some minutes. This led to a loss of about 121 billion dollars
+for S&P 500 and its related companies [11].
+I structure the attack plan in six stages:
+1) Finding an event/topic to build the firestorm attack
+on. This can be a past event or an error that the firm
+has committed in the past, which will be used as a basis
+for the upcoming attack. I define this event as the target
+topic.
+2) Using bots to create or amplify the latent state. By
+leveraging a botnet, an adversary can create a high num-
+ber of posts on social media, allowing the target topic
+to reach more people and giving them the opportunity
+to react negatively. This can eventually lead to a state
+where real people start to autonomously talk about the
+subject and begin to spread information about the target
+topic on their own. To facilitate this, the attacker can
+reuse an old trending hashtag or create a new one: the
+hashtag is the keyword to incite social action due to the
+information symbolised by the word itself.
+3) Letting the topic spread among people. The ideal
+situation for the attacker is that real people begin posting
+about the target topic, after learning about it from the
+botnet’s posts. This will bring more attention to the
+topic, possibly making it a trending one. For example,
+Twitter allows users to check what topics and hashtags
+are currently popular. If this happens, there will be
+moment in which there are enough people posting about
+the target topic, so that the firestorm can sustain itself for
+days, without any other post coming from the attacker’s
+botnet. I call this moment the fire point.2 Instead, if real
+people did not react negatively to the topic, or the topic
+did not reach enough people to allow the firestorm to
+reach the fire point, the discussion on the topic will slow
+down and will eventually end. In this case, I say that
+the firestorm is extinguished. However, the attacker can
+change the target topic and restart from Stage 1.
+4) Identifying human targets. Managers (e.g., Chief Tech-
+nical Officers, Chief Executive Officers) are the decision
+makers of a company. The attacker might want to keep
+a list of these people in order to use these names when
+the attack will move over from the company’s social
+network profiles to the employees’ ones. Identifying the
+people who are most proud to work for the attacked
+company can also be helpful in exerting more pressure
+on the company (since they have more to do with the
+value of the company).
+5) Focusing on workers. During the peak activity of the
+firestorm, those same bots that built the latent state will
+move their focus on the public social media profiles
+owned by employees of the attacked firm. These pro-
+files were identified in the previous step of the attack.
+This may cause the attention of the firestorm to shift
+towards the employees, also causing them to experience
+discomfort. Because the brand is usually at the center
+of the firestorm, focusing people will have a stronger
+impact on them, and it can disrupt internal processes.
+6) Performing the cyber attack. Because people will put
+less attention in following internal procedures, many
+safety best-practices adopted by the company may not be
+followed properly, or may even be ignored. The attacker
+can exploit this behaviour to their own advantage.
+In order to shift the focus from the company to the worker,
+it is necessary to optimise the timescale and timing of the
+transition, as it is not linear for people to attack the worker,
+but it can happen more easily if the negative event is of high
+negative impact and value. Shifting the attack on employees
+2In chemistry, the fire point is the lowest temperature at which a certain
+fuel will continue to burn for a minimum of five seconds, when ignited.
+
+has another side-effect, which is beneficial to the attacker:
+the organisations that are responsible for the public cyber
+security in every country cannot see the Firestorm attack
+on the company page, because the Firestorm is focused on
+workers only Such organisations will hardly be able to detect
+all comments and posts focused on workers, allowing the
+attacker to create a smoky form of the attack, which can
+bypasses conventional security measures, procedures and
+strategies. Since they have to focus primarily on the company
+under attack, therefore, possibly not give so much attention
+to analysing every single interaction against all the operators
+of the attacked company.
+III. BUSINESS SOCIAL MOOD-DISEASE AND NETWORK
+STRATEGY
+The Cambridge Analytica case highlighted the role and the
+importance of social media for the majority of the population
+and organisations. A document produced by the American
+Ministry of Justice, to examine the possible foreign influence
+on US, showed how there actually exist organisations (such as
+the IRA - Internet Research Agency) [36] that aim to influence
+individuals, public and private organisations [29].
+A great part of what is needed to successfully influence
+people lies to understand the initial conditions of the system,
+i.e. in the correct profiling of such people through data
+obtained on social networks. People who are more sensitive
+to certain issues, and those key people who can influence the
+most the community where they live and work are the main
+focused people for a social attack, because they have a central
+role (hubs) in the network.
+Profiling consists in obtaining (through a process of data
+collection and subsequent processing) an absolute or almost
+absolute understanding of a group of individuals or a single
+person, comprehending their habits and preferences [13]. The
+information obtained concerns political, musical and social in-
+terests, including the identification of their network of friends,
+colleagues, and much more. This information allow a much
+easier conveying of any content, as it is possible to understand
+who is most susceptible and interested on various topics,
+affecting their weaknesses, fears and interests. Furthermore,
+it is possible to infer who could possibly propagate a certain
+content through their network, exponentially increasing the
+chance of success if the subject in question is a person with
+an important or main role.
+Cambridge Analytica used the OCEAN model, related to
+personality traits, to understand preferences of many people
+in the US during the national election on 2016 [36]. The
+OCEAN model allows to send specific messages and contents
+to people who are sensible to a certain topic. This method
+is very different from the classic and standard mass commu-
+nication, because it is possible to send the right content to
+the right person. Unfortunately, the CA scandal was defined
+as classic political influence, the old-fashioned way, thus
+including prostitution, favouritism, etc. In reality, the scandal
+found “a new type of weapon” as Brittany Kaiser (former CA
+business development director) said during her question time
+(on Commons culture committee in 2018) to describe the work
+done from CA, but also to categorize AI as a real soft-power
+weapon [13].
+However, understanding hot topics for workers is not
+enough – in order to modify their mood and obtain a good
+social attack, a subject topic needs to be found as well.
+On social networks, during firestorms , people are usually
+triggered by three kinds of errors [15]:
+1) Social failure
+2) Communication failure
+3) Product or service failure
+Although they may seem similar, different types of events
+can lead to different types of dynamics and reactions. In the
+case of product or service failures, for example, performance-
+related crises raise doubts about the brand’s ability to deliver
+basic functional performance [9]. Another research has also
+identified not only short-term effects to a brand after a
+firestorm, but also measured long-term ones, at least two years
+after the latest firestorm [15].
+I hereby give an example for each of the aforementioned
+triggering factors.
+1) Social failure. The firm might be an accomplice of some
+accident or crime, like Nike with children shoes [10],
+[30] or the ING-DiBa case in 2012 [31].
+2) Communication failure. The firm might fail to commu-
+nicate properly, for example making negative comments
+regarding a certain community or movement [27].
+3) Product or service failure. The firm might distribute
+a product that harms consumers, for example a vaccine
+that can kill people [19].
+These failures and the firestorm stemming from them might
+cause affected employees to experience discomfort and panic,
+because coworkers, friends and other people in their net-
+work might see affected employees as the root-cause of the
+Firestorm.
+The social-cyber attack also provokes unlikely passive con-
+sequences for companies:
+1) The value of the company on the financial market could
+rapidly decrease; [11]
+2) People who worked in the company during the firestorm
+might be subject to discrimination in future, especially if
+the firestorm was caused by a (supposedly) unacceptable
+mistake that could have been avoided [26], [38].
+3) As the people, also the offended brand could carry a
+long-term stigma that would motivate other companies
+to make job offers to the personnel of the attacked firm.
+This could put it on an even greater disadvantage, as
+workers would be incentivized to leave the attacked
+company and accept the new offer.
+The network, as well as the importance and scope of the
+news, can thoughtfully influence the reaction and dynamics
+of the company. The network, as well as the importance and
+scope of the news, can thoughtfully influence the reaction and
+dynamics of the company. For example, when a company’s
+
+workers receive an high importance news, they may behave
+helplessly in relation to the importance of the news; feeling
+relieved of responsibility, since the event is bigger than their
+actions, they tend to pass much of the responsibility on to the
+company’s managers.
+Indeed, in times of disorder or chaos, Entropy increases with
+decreasing order, and emergency increases with increasing
+order: this happens because people within the organisation
+understood the emergency, and the organisation improve them-
+self to respond to it [39].
+When many workers in the company are panicking, the
+organisation’s CCO (Chief Communication Officer) will elab-
+orate and react to Firestorm on company pages, however, this
+cannot stop the social attack on the individual profiles of the
+employees. Hence, even people who are in charge of running
+communication processes and managers can panic, as the more
+is the duration of the firestorm, the higher is the chance of
+losing clients and reputation. This is a terrible situation for
+any company, especially after many years of work. However,
+managers are considered "critical workers" on the organisation
+chart, hence, they cannot be influenced by social manipulations
+and social diseases, because of the responsibilities they have in
+the company. While during the last century such organization
+charts had the form of a pyramid, usually with the CEO on the
+top, nowadays the AGILE model allows companies to organise
+their personnel in different ways within their organization
+charts. However, the legal and personal responsibility for every
+error or critical issue will be always be of the top manager
+of that area – for example, the CISO (Chief Information
+Security Officer) is usually responsible for the cyber security.
+A network side strategy can hard-influence workers close to
+managers and directors, contaminating directly the mood of
+the team, including the manager. In a more specific way, the
+attacker the hub from the company network, defusing also
+other workers from the company.Once the social-disease is
+already widespread on the company, and many people are
+stressed about the firestorm, the cyber attack can begin.
+IV. ASSESSING THE ATTACK SURFACE
+In this section, I introduce the possible actions that the
+adversary (or the real people that contribute to firestorm)
+can perform to further disrupt the target company’s business
+processes, to sink its corporate image, or to get classified
+information. To do so, I introduce a novel classification of
+these actions and analyze their impact on the fundamental
+properties of information security, that is, Confidentiality,
+Integrity and Availability [34].
+I show these actions can be divided in three categories:
+1) Controlling Large Scale Entities, that is, thousands
+or even millions of different actors performing several
+concurrent actions against a firm. These actors can act
+both remotely and physically, and can be both robots
+and humans.
+2) Leveraging Internal People, namely, exploiting mis-
+takes performed by employees (e.g., because they are
+stressed due to the firestorm), or having an insider threat
+who can extract classified information.
+3) Asking for Ransoms, that is, the adversary may want
+to ask for a payment to stop the firestorm. This would
+cause the bots to be shutdown, or even to defend the
+company on social medias.
+I hereby analyse the different actions within each category
+and their impact. This analysis is summarised in Table I.
+A. Controlling Large Scale Entities
+a) Denial of Service (DoS) Attacks: The adversary might
+want to harm the firm’s reputation by negating the availability
+of the services it offers. To this avail, the attacker can leverage
+botnets to send a very high number of requests per second to
+the target service, overwhelming the server and resulting in the
+service going down. If possible, the attacker could even reuse
+the botnet used to create the latent state, and rearm it with a
+DoS script. Alternatively, if the adversary is not a single entity
+but a large group of organised people, a DoS attack can be
+performed with simple scripts, without leveraging any botnet,
+as the large number of adversaries could be able to generate the
+traffic required to overload the server. In this case, however, the
+adversaries would have to carefully time their attack, and they
+might want to hide their location, for example by using a VPN.
+Finally, the adversary could encourage real people to overload
+the target firm’s servers, as they could co-ordinate the attack
+by using the bot profiles used for the hashtag propaganda.
+b) Physical Actions: Business processes can be also
+interrupted or slowed by legal, yet harmful, physical actions.
+One example is a demonstration around the firm’s premises:
+employees might not get to their workplace in time because
+people manifesting outside the building are blocking or slow-
+ing access to the premises, or they are creating more traffic
+than usual on the way to the building. Another example is
+people calling the organisation’s call centers with the only
+goal of protesting.
+B. Leveraging Internal People
+a) Human Error: Even though it is widely known that
+human error is one of the most prominent causes of security
+incidents [16], [43], most companies still do not adequately
+invest in training for their personnel, resulting in data breaches
+or other security related events [22]. This means that, if the
+attacker wants to obtain an initial foothold on the target
+organization’s systems, they might be able to do so without
+needing a firestorm attack, depending on the employees’ abil-
+ity of recognizing phishing emails or scam websites. However,
+workers who are experiencing firestorm, be it on the company
+they are working with or on their own profile, will be more
+inclined to break internal policies, hence committing mistakes,
+due to the perceived crisis [2].
+b) Offering Help: During the firestorm’s peak activity,
+the adversary itself contacts the attacked firm, pretending to be
+a professional (e.g, a consultant) who can help in mitigating
+the effects of the firestorm, for example as a Social Media
+Manager who has dealt with Firestorms before. This can
+
+happen via emails, social networks or through the corporate’s
+website, for example if the firm has some job openings and the
+adversary pretends to be a candidate. For smaller enterprises,
+the adversary may even show up in person to the attacked
+company’s premises. If the attacker manages to get hired, they
+might get access to classified information. I argue the attacker
+does not want to tamper with documents or attack the firm’s
+infrastructure while being an employee themselves.
+c) Insider Threats: Instead of joining the firm them-
+selves, the adversary might establish a contact with employees
+who are still in the attacked company but are not showing
+support on social media, or even manifested dissatisfaction
+towards the company. The attacker might want to try to
+persuade them in sharing confidential information, making
+them insider threats [25] – if they have success, not only they
+acquire classified information, but if the stolen content is also
+compromising for the firm, it could be published online to
+damage the firm’s reputation even more.
+C. Asking for Ransoms
+a) Extortion to Stop the Attack: The adversary contacts
+the attacked firm and proves the botnet that is performing the
+firestorm is in their control. They then ask for an arbitrary
+amount of money in Bitcoins to shutdown the bots, stopping
+a (hopefully) substantial part of the attack. In fact, if the
+firestorm already managed to incite many people in joining the
+social attack, the shutdown of the botnet might not stop or slow
+down the firestorm. If the adversary plans to attack multiple
+firms with their firestorms, they to avoid situations like this,
+because the odds of a victim paying a ransom is proportional
+to the reliability of the attacker in stopping the attack once
+they receive the money. In other words, the attacker must be
+considered “trusted” in stopping the attack if the ransom is
+paid, so victims are more incentivized to pay [4].
+b) Defence as a Service: The adversary contacts the
+attacked firm, but instead of showing they are in charge of
+running the attack and asking money to stop it, they try
+to sell a fire(storm)fighter service to the victim, supposedly
+consisting on bots defending the reputation of the firm: this
+is basically a reversed firestorm, in which those same bots
+that built the latent state now defend the company: to avoid
+drawing excessive attention, the attacker might slowly change
+the proportion of attacking bots versus defending ones, until
+they are all defending the company.
+V. CASE STUDY: CD PROJEKT RED
+On December 10, 2020, CD PROJEKT RED released a long
+awaited game called Cyberpunk 2077. This game was very
+popular even before its release and it generated continuous
+social hype from the video game community throughout its
+development, also winning the “Best Game Awaited” from
+Golden Joystick Awards for two consecutive years. [42] As
+shown on Figure 1 and Figure 2, hype for the game substan-
+tially increased during the 10 days before the release of the
+game, reaching its apex on December 10, when the hashtag
+#Cyberpunk2077 was tweeted 193,900 times on Twitter,
+TABLE I
+SOCIAL ATTACK SURFACE ASSESSMENT
+Category
+Action
+Impacts
+Confid.
+Integ.
+Avail.
+Rep.
+Large Scale
+DoS Attack
+No
+No
+Yes
+Yes
+Phys. Actions
+No
+No
+Yes
+Yes
+Internal People
+Human Error
+Yes
+Yes
+Yes
+Yes
+Help Offer
+Yes
+No
+No
+No
+Insider Threat
+Yes
+No
+No
+Yes
+Ransoms
+Extortion
+No
+No
+No
+No
+Defence Service
+No
+No
+No
+No
+Confid.: The action can affect the Confidentiality property. | Integ.: The
+action can affect the Integrity property. | Avail.: The action can affect the
+Availability property. | Rep.: The action can negatively affect the reputation
+of the company.
+from users of 53 different nationalities. During this time span,
+many other hashtags regarding the game were very popular,
+for example #Cyberpunk2077Hype was retweeted 10,000
+times [41].
+However, a few days after the release , the Cyberpunk 2077
+topic arise again, this time associated with queries related to
+patches and refunds. In fact, the game was released too early
+and many bugs were present: due to this, several people had
+asked a refund to CD PROJEKT RED, often also writing
+a bad review for the game on online stores. This created a
+"information-disease" within the company, just like the one
+described in Section III: in this case, CD PROJEKT RED’s
+employees became stressed and felt pressure related to the
+quality of Cyberpunk 2077, in which they had invested more
+than two years of hard work. [42]
+In early February 2021, only 60 days after the game’s
+release, CD PROJECT RED was hit by a ransomware attack
+and attackers were able to extract the source code of several
+games, including administrative files [8]. The attackers then
+threatened the company of leaking or selling the stolen code
+and files, unless the firm paid a large amount of money to the
+cyber-criminals. In the end, CD PROJECT RED refused to
+negotiate with the attackers, stating on a press release that they
+would “not give in to demands or negotiate with the actor”,
+also confirming that no personal information was obtained in
+the attack and that they were working with law enforcement to
+track down the attackers [7], [35]. Later on, security analysts
+found the stolen source code while being auctioned on the dark
+web for a minimum price of 1 million USD. [40] The auction
+was closed after the attackers stated they had received an offer
+that satisfied them [40] Within a week of these auctions, the
+code was shared online via social media, and CD PROJECT
+RED began using DMCA take down notices to remove posts
+containing their code [24].
+The social hype that CD PROJEKT RED generated for
+Cyberpunk 2077, was used by hackers to threaten the company
+in order to extorting money, but also, had a side effect,
+i.e. damaging the company’s reputation, that can bring to
+undermine the sales of other long awaited games.
+In Table II I show the results of the sentiment analy-
+
+sis, obtained from tweets and comments for the hashtag
+#CDprojectRED. Data collected from Twitter respects the
+timeline of Cyberpunk 2077’s release and its development;
+data shown in the table can be organised in three categories:
+before release (October and November), during release (De-
+cember and January) and after the release of Cyberpunk 2077
+(February).
+It is possible to observe that in October and November the
+sentiment remained neutral-positive with a few oscillations. In
+December, when the game was released, I can observe a small
+increase in the negative sentiment due to the high number of
+bugs present in the game, however, this increment is quite
+negligible. In January, when a greater number of players were
+playing the game, the negative sentiment became stronger than
+the positive one, causing not only a negative compound (-
+0.111), but also a neutral-negative sentiment for the game and
+for the developers. Finally, on February the sentiment returned
+neutral overall, however, the presence of negative sentiment is
+still stronger compered to the one in October and November.
+These data show how much pressure the CD PROJEKT
+RED company had to experience during the release of the
+game. Additionally, in Figure 3, I show the financial value
+of the company during the whole game release timeline, also
+marking the two critical events that occurred: the yellow line
+indicates the release of the game, while the red line indicates
+the ransomware attack. I can see that, after the release of the
+game, the financial value of the company suffered a sudden
+drop, that was likely conditioned by customers losing trust in
+the company due to the presence of many bugs in the game,
+bad reviews and critics. I can see that the company regains
+more than half the value lost during the next two months,
+however, the ransomware attack causes another drop in the
+financial value of the company due to customers losing trust
+in the company again, this time from a security perspective.
+TABLE II
+VADER SENTIMENT ON #CYBERPUNK2077 FROM TWITTER
+Months
+Negative
+Neutral
+Positive
+Compound
+October
+0,085
+0,757
+0,150
+0,163
+November
+0,079
+0,766
+0,149
+0,163
+December
+0,087
+0,750
+0,161
+0,153
+January
+0,143
+0,758
+0,093
+-0,111
+February
+0,104
+0,745
+0,145
+0,120
+VI. BUSINESS DEFENCE STRATEGY
+To avoid dangerous events for companies, human factor is
+a crucial element [37], however it is also possible to create
+specific defence strategies. Failures introduced in Section III,
+i.e. social failures, communication failures and product or
+service failures can be analysed to prevent incidents. To the
+most of us, the news that a particular piece of information (e.g.
+a meme, a hashtag) went “viral”, reaching millions of nodes
+in a short period of time may seem purely random and hence
+unpredictable, but Kolli et al. [21] discovered that, at least 20%
+of the times, the cascade volume changes in a manner that
+appears to be random, and in the remaining 80% it is possible
+Fig. 1. Interest Score showing social hype for the release of Cyberpunk 2077
+Fig. 2. Queries showing social hype for the release of Cyberpunk 2077
+to predict the cascade’s future volume. Hence, it is possible
+to create short-term strategies to detect firestorm attacks while
+they are still in the early stages, i.e. while the latent state is
+being built. However, it is also possible to create long-term
+defence strategies with a proactive governance. A possible
+proactive strategy for the long-term could be as follows:
+1) Organise internal company procedures to help employ-
+ees protect themselves against various attacks on social
+media (like Linkedin);
+2) Organise procedures outside the company, such as con-
+Fig. 3. Financial value of CD PROJEKT RED and critical events
+
+120
+100
+ score
+80
+Internet search s
+60
+40
+20
+0
+01/11/2020
+03/11/2020
+/11/2020
+07/11/2020
+09/11/2020
+11/11/2020
+13/11/2020
+15/11/2020
+17/11/2020
+/2020
+21/11/2020
+23/11/2020
+25/11/2020
+27/11/2020
+29/11/2020
+01/12/2020
+/12/2020
+05/12/2020
+07/12/2020
+09/12/2020
+19/11/
+03/20
+18
+Query search score
+16
+14
+12
+10
+8
+６
+4
+2
+0
+date
+ date500
+450
+400
+350
+Financial value
+300
+250
+200
+150
+100
+50
+Development
+Game release
+Ransomwere attack
+0
+01.10.2020
+07.10.2020
+13.10.2020
+19.10.2020
+23.10.2020
+29.10.2020
+04.11.2020
+10.11.2020
+17.11.2020
+23.11.2020
+27.11.2020
+03.12.2020
+09.12.2020
+15.12.2020
+21.12.2020
+29.12.2020
+07.01.2021
+13.01.2021
+19.01.2021
+25.01.2021
+29.01.2021
+04.02.2021
+10.02.2021
+16.02.2021
+22.02.2021
+26.02.2021
+04.03.2021
+10.03.2021
+16.03.2021
+22.03.2021
+26.03.2021tacting allied/partner companies for help with the various
+attacks on social media;
+3) Create in advance supporting bots that will defend the
+company automatically;
+4) Create an international database of accounts that have
+made firestorm. The database, accessible to all organi-
+sations, both public and private, will help to understand
+whether the type of firestorm taking place is real or
+artificially created. [12]
+These three possible actions can be highlighted by the
+mass media, which will publicly show that the firestorm is
+being fought because other people or organisations began
+defending the attacked company. Hence, these actions allow
+the firestorms to calm down, and eventually to be extinguished,
+faster than simply doing nothing. [15] If a company has done
+something enormously wrong in the past, it is possible that
+every time the same company does something wrong, there
+is a chance that another firestorm can restart, either for the
+recent event or also for the past one. In fact, the firestorm can
+come back with an interval of about 2 years [15].
+In case of social failures, there is also an additional side-
+effect that must be mitigated, that is, the firestorm naturally
+expands to the employees without the manipulation of the
+adversary. Example defence strategies against this side-effect
+could be implemented as follows:
+1) Let people from outside and inside the company on
+social network, dialogue about that topic (such as the
+case of carnivores vs vegetarians at ING-DiBa [31]).
+This strategy can increases the number of followers;
+2) Blame an entity that is external to the company as a
+scapegoat, so the Firestorm can move from the company
+to the designed entity. Even if it is not very moral, it is
+something that usually works;
+3) Depending on the strength, length, and breadth of the
+attack, it is possible to make strategy about possible
+reaction for company.
+a) Social failure: If the firestorm is linked to a partner
+company, or only a certain sector of the company
+is under attack, immediately distance yourself from
+them.
+b) Communication failure: The goal here is to safe-
+guard the company’s reputation and authority. In
+this case, try to detach yourself immediately from
+the communication error, and continue with the
+company’s reputations strategy, making it appear
+that it was just an accident on the road. Further-
+more, apologising for the event never hurts.
+c) Product or service failure: Instantly block the pro-
+duction of the affected product or the provision
+of the service. Organise a commission that can
+evaluate the quality of product/service. Even if it is
+complicated given the amount of partners, quality
+standards and corporate continuity, this action, if
+done in time, creates a good defensive shield at
+the communication level, as people can understand
+that the company itself has also understood the
+problem, limiting the damage;
+Timing is essential during Firestorms, first of all to
+understand whether the type of firestorm is real or artificial
+(you can tell by the date of creation of the accounts that do
+firestorm – if the initial accounts were born recently, they
+are probably bots, hence artificial); secondly for improving
+the cyber defence and be prepared for a possible cyber
+attack; tertiary for the public reaction, because it means
+that the affected company has noticed the failure faster or
+as fast as other people (who are doing the firestorm on
+social networks) and will promptly react to the problem,
+reassuring customers that it will be solved. This will help in
+calming down or extinguishing the firestorm. For example,
+the carnivores vs vegetarians case at ING-DiBa was caused
+by a communication failure. The company had never had so
+much traffic on its Facebook page before, and they saw in this
+an opportunity to increase the number of their followers. In
+fact, after a few days had passed from the firestorm, and the
+attackers were still posting, newly-acquired followers jumped
+into the debate and started defending the company. [31]
+Obviously,
+depending
+on
+the
+type
+of
+firestorm,real
+or
+artificial, it is necessary for the company to adapt its
+strategies according to the type of attack (real or artificial).
+The prevention part, of course, works in both cases, but
+understanding who you are fighting against and the causes,
+helps to save the reputation of the company, and sometimes
+even the company itself.
+VII. FUTURE WORK
+In one of the next jobs, I would like to implement different
+pressure dynamics, i.e., either implement rapid, massive, and
+incisive firestorms, or permanent, with few accounts firestorm.
+Depending on the firestorm, these types of dynamics can
+change the pressure on companies and workers in different
+ways, perhaps showing that for some companies it is better
+to have a permanent firestorm, or for others a rapid one.
+Another aspect I would like to draw attention to in future work
+is also how people are contacted in the company, i.e. with
+messages that are more likely to provoke an ethical reaction,
+for example, when people are contacted by bots and they point
+out to the worker the disaster he has made to his company.
+This case is very interesting, as it is possible, after ’moralising’
+the worker, to apply social engineering strategies to facilitate
+the cyber attack. On the other hand, on the side outside the
+company, i.e. not focused on employees, strategies can be used
+to increase the chance of a successful cyber attack, or extortion
+of information or money. For instance, during the firestorm,
+it is possible to contact the company under attack, and pose
+as the national cyber security agency, initiating strategies such
+as:
+1) Passing themselves off as the national cyber security
+agency, they say that most are fake accounts and get
+information on their security;
+
+2) Passing themselves off as the national cyber security
+agency, enter in their computer system.
+3) Passing themselves off as the national cyber security
+agency, saying they are carrying out a cyber attack
+to test their cyber defences, carry out a second attack
+immediately afterwards, exploiting the information from
+the first attack and passing on part of the defences, or,
+say they are not defending themselves against the first
+attack so as to obtain the desired data.
+In any case, these kinds of interactions will be carried out
+by means of computer simulations, since for obvious ethical
+reasons it is impossible if not extremely difficult to apply these
+strategies.
+VIII. CONCLUSIONS
+In this paper, I have shown how some events related to
+cyber security are linked to certain social dynamics. When
+social dynamics are mixed and linked to cyber purposes,
+classic attack types (cyber or social attack) can no longer be
+defined, but social-cyber attacks, as the effectiveness of one
+also induces a probability of success of the other.
+I introduce an novel model allowing researchers and com-
+panies to (1) understand when companies and organisations
+have fragile defence against a social-cyber attack, (2) illustrate
+how company and organisation can defence them self from
+firestorm, (3) proving that social-cyber attack must be defined
+as a possible high risk event as multi domain sector, and (4)
+showing a now model of cyber attack, with a multidisciplinary
+sociological approach to increase the potentiality of common
+cyber attack. The data collected from CD project red’s event
+case, shows how these types of attacks, although still little
+known, may become a norm in the future, as the company’s
+assets are not only its human capital, or the production of
+goods and/or services, but also its own reputation.
+IX. AUTHORS & PAPER INFORMATION
+A. Data gathering
+I collect tweets related to the topics #Cyberpunk2077 by
+using Tweepy and the Twitter archive API. Both service use
+the permission from Twitter to obtain and gather data, but
+any downloaded topic need revisions and cleaning process to
+increase the quality of the research. For example, I found
+many copy-paste tweets (caused by spamming process, or
+fake-account/bot), and also several tweets had (during the
+Vader Sentiment Analysis) incomprehensible word for the
+Vader program, and I deleted it. For any topic I use the
+same methodology to obtained standard and quality data. In
+addition, to obtain the correct amount of tweet (define as the
+number of tweet) for each day/hour I use getdaytrends.com, a
+specific site where it is possible to monitoring every topic in
+real-time and also aged topic. In total, our data count more then
+∼5000 Tweet. I obtain the Financial data of CD project RED
+from https://www.investing.com/equities/cdproject-historical-
+data site.
+B. Author Contributions
+Investigation and data resources, methodology, data cleaning
+and software, A.R.; All authors have read and agreed to the
+published version of the manuscript.
+C. Funding
+The author(s) disclosed receipt of the following financial
+support for the research, authorship, and/ or publication of this
+article: This project has received funding from the University
+of Catania.
+D. Author biographies
+Andrea Russo is a PhD candidate in Complex Systems
+at the University of Catania. He is currently working at the
+Department of Physics and Astronomy. He collaborated with
+CNR Ibam, he also has worked purely on projects involving
+technology and society.
+His
+main
+research
+field
+and
+interests
+are
+focused
+on
+the study and the development of Computational social
+method to explain social complexity, in particular field like
+Politics - Economics - Business and Defense-Security sector
+applications.
+Orchid: 0000-0003-3816-0539
+Corresponding author. Email: Andrea.russo@phd.unict.it
+I would like to thank "Vereos" and "Andrea metal clone",
+who helped me in idealising and refining the paper.
+REFERENCES
+[1] E. B. Alan Bryman. Business research methods, 2nd ed.oxford: Oxford
+university. Oxford University Press, 2007.
+[2] L. Bakos, D. D. Dumitras,cu, and K. Harangus.
+Human factor pre-
+paredness for decentralized crisis management and communication in
+cyber-physical systems. Sustainability, 11(23):6676, 2019.
+[3] G. Carrer and F. Bechis. Così la cina fa propaganda in italia, con i bot.
+ecco l’analisi su twitter di alkemy per formiche. Formichiere.it, page 1,
+2020.
+[4] E. Cartwright, J. Hernandez Castro, and A. Cartwright.
+To pay or
+not: game theoretic models of ransomware. Journal of Cybersecurity,
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+bbc.com, 2021.
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+“il ruolo della cyber threat intelligence nelle
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+SOCINT Press, 2021.
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+The effect of stress and satisfaction
+on productivity. International Journal of Productivity and Performance
+Management, 2010.
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+digital age: The short-and long-term effects of social media firestorms on
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+35(4):557–574, 2018.
+
+[16] K. Hughes-Lartey, M. Li, F. E. Botchey, and Z. Qin. Human factor,
+a critical weak point in the information security of an organization’s
+internet of things. Heliyon, 7(3):e06522, 2021.
+[17] iso.org. Iso 9001:2015. iso.org, 2015.
+[18] iso.org. Iso/iec 27002:2022. iso.org, 2022.
+[19] A. G. Kate Connolly and J. Henley. Chaos in germany and italy after
+suspension of oxford vaccine. The Guardian, 2021.
+[20] R. Knight. If your company is going through a public scandal, should
+you leave? Harvard Business review, page 1, 2018.
+[21] N. Kolli, N. Balakrishnan, and K. Ramakrishnan. On quantifying pre-
+dictability in online social media cascades using entropy. In Proceedings
+of the 2017 IEEE/ACM International Conference on Advances in Social
+Networks Analysis and Mining 2017, pages 109–114, 2017.
+[22] P. Langlois. 2020 data breach investigations report, 2020.
+[23] J. Lehtonen. Kriisiviestintä. Mainostajien liitto, 1999.
+[24] Lorenzo. Cd projekt red uses dmca to take down tweets sharing stolen
+game code, 2022.
+[25] G. Mazzarolo and A. D. Jurcut. Insider threats in cyber security: The
+enemy within the gates, 2019.
+[26] K. McLeod. Workers left destitute after hes scandal say bosses had cash
+to pay wages but refused. dailyrecord, page 1, 2019.
+[27] M. Monkey. Twitter users not lovin’ mcdonald’s. The Guardian, 2012.
+[28] K. Nuortimo, E. Karvonen, and J. Härkönen. Establishing social media
+firestorm scale via large dataset media analytics. Journal of Marketing
+Analytics, pages 1–10, 2020.
+[29] U. D. of Justice. Report on the investigation into russian interference in
+the 2016 presidential election. Department of justice, 2019.
+[30] J. Oliver. Learning the lessons of brent spar saga. Politico, 1995.
+[31] J. Pfeffer, T. Zorbach, and K. M. Carley.
+Understanding online
+firestorms: Negative word-of-mouth dynamics in social media networks.
+Journal of Marketing Communications, 20(1-2):117–128, 2014.
+[32] F. M. Rinaldi, G. Giuffrida, and T. Negrete. Real-time monitoring and
+evaluation-emerging news as predictive process using big data-based
+approach, 2017.
+[33] R. R. Riverso.
+Barcellona, arrestato l’ex presidente Bartomeu, Mar.
+2021.
+[34] S. Samonas and D. Coss. The cia strikes back: Redefining confidentiality,
+integrity and availability in security.
+Journal of Information System
+Security, 10(3), 2014.
+[35] J. Schreier.
+Cd projekt ransomware hack severely disrupts work on
+cyberpunk updates. bloomberg.com, 2021.
+[36] I. senate USA. Background to “assessing russian activities and inten-
+tions in recent us elections”: The analytic process and cyber incident
+attribution. USA Senate, 2017.
+[37] C. Simonelli. Prima educare, poi comprare. il fattore umano nella lotta
+al ransomware. Formiche.net, page 1, 29 maggio 2021.
+[38] K. Strauss. How volkswagen rallied its employees after its emissions
+scandal (at least for now). Forbes, page 1, 2017.
+[39] M. Tang and X. Mao. Information entropy-based metrics for measuring
+emergences in artificial societies. Entropy, 16(8):4583–4602, 2014.
+[40] Topic. Cd projekt red source code reportedly sells for millions in dark
+web auction [updated] | ars technica, 2022.
+[41] A. User. #cyberpunk2077hype • united states • twitter trending hashtag,
+2022.
+[42] Wikipedia contributors.
+Cyberpunk 2077.
+https://it.wikipedia.org/w/
+index.php?title=Cyberpunk_2077&oldid=130410919.
+Accessed: NA-
+NA-NA.
+[43] C. C. Wood and W. W. Banks.
+Human error: An overlooked but
+significant information security problem. Comput. Secur., 12(1):51–60,
+feb 1993.
+

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+ 
+ 
+ 
+ 
+Femtosecond Laser Engraved 2D Tunable Optofluidic Liquid 
+Core/Air Cladding Channel Waveguides on PDMS 
+Sanyogita*, Amar Ghar  and P. K. Panigrahi 
+Centre for Lasers and Photonics, Indian Institute of Technology, Kanpur-208016 (UP). 
+sanyogita.iitk@gmail.com 
+ 
+We have demonstrated fabrication and characterization of 2D liquid based multimode optical waveguide structures over 
+Polydimethylsiloxane (PDMS) material based chip. Fabrication of two separate microsturures, one with width of 14 micron 
+and depth of 27 micron while the other with width as well as depth of  110 micron,  was achieved by femtosecond laser 
+micromachining process. The dye solution is passed through the microstructure from one end to the other; wherein dye 
+solution acts as the core while PDMS and air act as cladding medium. The femtosecond laser micromachining parameters 
+are optimized in terms of laser power, pulse width, writing speed, focused beam size etc. Quality of fabricated 
+microstructures is confirmed by microscopic analysis. The confirmation of liquid core/air cladding based waveguide is 
+obtained through the spectral and modal analysis. The optical analysis has been done by using fluorescence light coupled 
+out from waveguide structures filled with different dye solutions. These waveguide structures give strong light 
+confinement and intense interaction between dye solution and pump light. The developed micro structures are tunable in 
+terms of intensity, wavelength and beam size. Such micro structures can be implemented in design and development of 
+lab-on-chip micro lasers and sensing applications in any multifunction lab-on-chip devices. 
+Introduction  
+Optofluidic is a great research platform where the advantages of 
+both optics and microfluidics can be combined in a single chip to 
+move towards highly compact, portable and multifunctional devices 
+[1]. This optofluidic lab-on-a-chip (LOC) approach provides a huge 
+potential in terms of low-cost optical sources, sensors, liquid-liquid 
+waveguide, liquid core waveguide and real time detection. 
+Particularly in photonic science, and more specifically in the micro 
+and nano regime, the integration of fluid and light in the same path 
+offers the capacity to reconfigure the device in accordance with the 
+choice of fluid opted as the fluid medium and thus providing 
+dynamic and powerful practical tuning mechanism, making it 
+customizable in real time [2, 3]. 
+Nonetheless, the fabrication and characterization process are 
+complicated owing to the miniscule dimensions of such 
+microstructures and managing the required smoothness at the 
+edges of microchannel and waveguide wall. High precision handling 
+of chip is also a must to minimize optical losses and for accurate 
+control over light and fluid in the micro/nano regime to maintain 
+good functionality. In the liquid core/air cladding waveguide chip, 
+the refractive index of core material has to be higher than that of 
+the cladding so as to enable total internal reflection (TIR) 
+phenomenon for the refractive index guided mode.   Moreover, dye 
+solutions with different host materials and concentrations have 
+broad range variation in refractive index to that of water. Such an 
+enhanced range helps in sustaining the liquid core-air waveguide 
+over the long flow path for a much higher operational time. This 
+feature provides for a substantial increase in wider applications of 
+mode for such type of optofluidic chip. 
+Optofluidic waveguides can confine light in small dimensions and 
+generate high intensity optical beam over a long distance, creating 
+a potential for tremendous applications in the field of 
+environmental monitoring, bio-sensing, analytical chemistry etc. [4]. 
+ 
+Various methods have been proposed to fabricate 2D structures; 
+among them, structure fabrication using soft lithography process is 
+widely prevalent [5,6]. But the soft lithography process in itself have 
+a number of disadvantages like involvement of multiple fabrication 
+steps, high rate of errors while achieving required depth of 
+microstructures, longer time of fabrication etc. Most noticeable 
+drawback of soft lithography is that it requires another lithography 
+method such as photolithography or e-beam lithography to 
+fabricate the stamp master used in further development process of 
+microstructure [6]. On the other hand, Femtosecond laser based 
+direct writing has many advantages over other conventional 
+methods such Excimer laser writing, CO2 laser writing-beam 
+lithography and soft lithography etc.[6,7] for fabrication of 
+microstructures. Femtosecond laser interaction with soft materials 
+has opened up a new field of waveguide fabrication methods for 
+structures on the surface as well as inside of transparent materials. 
+A femtosecond laser emits pulsed beams with durations of tens or 
+hundreds of femtosecond region which, nowadays, are used for 
+high-quality micro and nanofabrication. As the energy deposition 
+time of femtosecond laser is shorter than time required to release 
+the energy in the form of heat using electron-photon coupling 
+process, heat affected zone is completely suppressed during the 
+laser pulse interaction even with soft material like PDMS [7]. This 
+feature enables laser processing on PDMS with high precision and 
+resolution. Another advantage of femtosecond laser processing 
+over conventional methods is the capability of sculpturing complex 
+shapes at micro and nanoscale in transparent materials. With the 
+help of focused fs-laser beam one can achieve extremely high peak 
+intensity in the focused region which provides for high precision in 
+setting up interaction region at the surface or even inside the 
+volume. This feature not only eliminates a complicated and multiple 
+patterning processing, involved in the conventional methods like 
+photolithography for 2D fabrication, but also makes it feasible to 
+ 
+ 
+
+ 
+ 
+ 
+ 
+create complex 2D structures which were not easily achievable by 
+other conventional methods. The application of femtosecond 
+micromachining to develop the optofluidic devices improves their 
+structural and optical qualities to such an extent that it could 
+provide a major alternate platform to innovate and produce novel 
+optical devices on mass production level. Hence, this unique 
+technique is going to contribute as a promising tool in the photonics 
+fields and will help in emergence of new businesses once it reaches 
+commercialization. 
+In this paper, we have demonstrated the fabrication of micro 
+structures by using femtosecond direct writing along with 
+development of liquid core-based waveguide. Structuring of 2D 
+micro channels on the surface of PDMS is fabricated by f-s laser. 
+These microchannels are converted to a super hydrophobic nature 
+which can provide for an effective wave guiding. For light flow path, 
+R6G and RH101 dye solutions were selected as liquid core medium. 
+These dyes are distributed evenly along the length of the two 
+prototypes that we have fabricated as two microchannels. 
+Concentration of dye solution is chosen in such a way that 
+refractive index of liquid medium is slightly higher than that of 
+PDMS and air so that the PDMS and air ends up acting as a clad. 
+Cross sections of these waveguide systems were captured by a CCD 
+camera. Role of incident power, concentration of liquid dye and 
+photo bleaching have been successfully studied thereof. 
+Experimental Details 
+Femtosecond laser micromachining process has been used to 
+fabricate two distinct dimensioned microstructures, each on a 
+separate PDMS surfaces with a provision of inlet and outlet at the 
+terminal ends for flow of liquid across the microchannel. These 
+microchannel act as two unique liquid core/air clad waveguides. Fig. 
+1 shows the schematics of experimental set up for femtosecond 
+laser-based micromachining system. The proposed experiment 
+consists of regenerative Ti: Sapphire based amplified laser system 
+(CLRK-MXR, USA) capable of delivering a maximum output power of 
+800 mW with pulse width of 120 fs having central wavelength of 
+775 nm and repetition rate of 1 KHz. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+Fig. 1: Femtosecond micromachining fabrication setup for 2D 
+Microstructures/hallow waveguide structure on PDMS 
+The output beam from fs-laser system is focused on surface of 
+PDMS sample using 10X objective lens and beam aligning system 
+(OPTEC Belgium). All the microstructures are created by successive 
+translator movements of PDMS sample mounted on micro-position 
+stage without any movement of focused laser beam. The PDMS 
+substrate is irradiated with focused laser beam. The key steps in the 
+experiment includes focusing lens and micro-position translation 
+stage with 1 um resolution as shown in Fig 1. The focusing objective 
+lenses are used to converge the laser beam providing a greater 
+depth of field and smaller spot size as per the calculated 
+requirement which is important for precision laser micro-machining 
+process. Micro-position stage is used to move the sample as per the 
+designed program.  The computer-controlled laser power and 
+micromachining system ensures that position errors and beam 
+distortions are minimized over the entire scan region.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig.  2: Schematics of: a. Waveguide-I cross section; b. Waveguide-
+II cross section 
+For this experimental study, two straight microchannels on 
+separate surfaces of PDMS have been fabricated successfully with 
+different focusing lens. Both the microchannels are fabricated with 
+different lasing power and focusing lenses. First microstructure 
+(larger microchannel) is fabricated with a width of 110 µm and a 
+depth of 110 µm and the second microstructure (smaller 
+microchannel) with a width of 14 um and a depth of 27.937 µm as 
+shown in Fig. 2. The larger microchannel has been fabricated by 
+setting the laser power at 25 mW with a spot size of 15 µm (writing 
+speed was kept 1mm/sec) and using multi-pass laser scan over the 
+square shaped cross section. Based on multimode waveguide, the 
+target cross-section is scanned 10 times horizontally and 5 times 
+vertically with a beam overlap of 10 µm. Fabrication of inlet and 
+outlet has also been done by fs-laser using multi-pass laser scan. 
+The smaller microchannel (waveguide I) as well, has been fabricated 
+with multi-pass laser scan but with slightly different writing 
+parameters. Here laser power was taken as 18 mW with a beam 
+spot size of 8 µm and horizontal scanning was done only twice with 
+a beam overlap of 6 µm (writing speed 1mm/sec).  After the 
+measurement width of channel was found to be 14 µm and depth 
+was 27.937 µm. In order to flow the dye solutions through 
+fabricated channels, uniform inlet and outlet connected to central 
+microchannels have also been fabricated with a multi-pass and 
+multi scan using fs-laser. Inlet as well as outlet for bigger 
+microchannels measure 110 µm in width and 40 µm in depth and 
+for smaller microchannel width was 110 µm and depth was 20 
+microns. In both the cases we have kept the depth of inlet and 
+outlet less than the central microchannel, for easy flow of liquid in 
+to it from. 
+ 
+Femtosecond Laser
+M1
+M2
+BS
+CCD
+Objective
+Lens
+Sample
+Comp. Controlled
+Translation Stage
+ 
+ 
+PDMS 
+Air 
+PDMS 
+Dye 
+14 μm 
+27 μm 
+ 
+ 
+PDMS 
+PDMS 
+Air 
+Dye 
+127 μm 
+127 μm 
+ 
+ 
+
+  
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+The corresponding width and depth of developed microstructures 
+have been confirmed by image analysis obtained with confocal 
+microscope (Olympus LEXT OLS 4000) as shown in Fig. 3 above. This 
+system capable of resolution up to 10 nm in Z direction and 120 nm 
+in X-Y plane. The super hydrophobic channels are effective in 
+creating air cladding between the dye filled liquid core and solid 
+walls of PDMS, thus providing a good coupling for TIR and the 
+waveguide. Here, due to 2D wave guiding, scattering and diffraction 
+of visible light still persists to the channel walls. Light undergoes TIR 
+at the front end of the channel too. Due to femtosecond structuring 
+on the PDMS material, the PDMS channel wall is also made 
+  
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+hydrophobic which controls the losses of waveguide. After 
+measuring the contact angle for femtosecond direct-written 2D 
+microchannel as shown in Fig. 4, the hydrophobicity was checked 
+for the contact surface modified due to exposure of femtosecond 
+laser with similar parameters that one used to fabricate 
+microstructures on PDMS respectively. It was found that channel 
+has been converted into a hydrophobic channel. These hydrophobic 
+channels have low solid fraction that can effectively support the 
+liquid-core/air cladding waveguide configuration on lab-on-chip 
+platform. Hence, this unique structure allows an effective control 
+and flow of light from one end to other.  
+ 
+ (a) 
+         
+(c) 
+ 
+(d) 
+ 
+ 
+ 
+(a) 
+(b) 
+Microchannel    (b) 
+Fig. 3: (a) 2D waveguide structure-I over PDMS, (b) Cross section of Waveguide structure-I (c) 2D microstructure-II 
+over PDMS (D) Cross section of microstructure-II 
+Fig. 4: Contact angle measurement for (a) Plane PDMS surface and (b) for PDMS surface exposed with femtosecond 
+laser 
+ 
+ 
+ 
+
+Obg
+Inlet
+480
+320
+160
+160
+320
+480
+640
+Microchannel112.3
+112.332
+64
+42
+/21
+96
+128
+96
+64
+128
+32Inlet
+08
+320
+160
+320
+480
+640
+160
+Microchannel320
+480
+640
+0
+160
+Microchannel 
+ 
+ 
+ 
+Implementation of microstructure as an optical 
+waveguide 
+The two fabricated microchannels, with 2D square and rectangle 
+shape cross section respectively are filled with liquid dye medium in 
+order to convert it into liquid based multimode waveguide 
+microstructures. The structures act as liquid-core waveguide 
+platform when the refractive index (n) of cladding material 
+(PDMS/air) is smaller than that of the flowing dye solution which 
+acts as the core and enable the total internal reflection for the 
+configuration of the index-guided mode [8, 9] 
+The waveguide losses are also sensitive to the roughness of the 
+surfaces of the waveguide walls. As the waveguide walls are pretty 
+smooth in case of femtosecond fabrication, the losses are very 
+much minimized in comparison to other conventional fabrication 
+methods. Other challenges and issues in these experiments are also 
+resolved as gas (i.e., air) is used as cladding material [9, 10]. Air has 
+a much lower refractive index (nair=1.0) than most of the solid and 
+liquid materials, thus it allows a wider range of incident angles. Air 
+also has much lower viscosity than that of any liquid so that it can 
+significantly reduce the hydrodynamic friction and Joule heating at 
+the interface between the core and the cladding [10]. Higher 
+refractive index difference between the liquid core and air cladding 
+(Δn= 0.407) helps to increase the amount of light trapped inside the 
+core and avoids the diffusional mixing problem normally observed 
+in liquid to liquid L2 waveguide.  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+In presented case, two types of dyes have been used as the gain 
+material to demonstrate the concept of liquid-air waveguide on a 
+chip. First dye is Rhodamine-6G dissolved in ethanol and benzyl 
+alcohol while the second one is Rhodamine-101 dissolved in 
+mixture of ethanol + benzyl alcohol in a concentration range of 
+1mM to 5mM for both liquid core solutions. The corresponding 
+change of refractive index of fluid observed by varying the dye 
+solution concentration for both dye solutions is measured by the 
+refractometer (Abbemat 500). The refractive index difference of 
+core and clad has been selected between 10-3 to 10-2 for index for 
+varying concentration form of R6G and Rh101 from 1% to 10 %. 
+From measurement, it is evident that dye solutions with different 
+concentration can act as two different liquid core medium with 
+varying characteristics. For example, in case of 1mM concentration 
+Rh-6G dye solution (n2=1.4030) in mix solution of (ethanol + benzyl 
+alcohol) is higher than that of cladding material i.e. air (n1= 1) and 
+PDMS (n3= 1.40). Liquid filled channel acts as a core in this case 
+wherein light propagates through liquid core waveguide by 
+satisfying condition of total internal reflection. This has been 
+demonstrated through the resulting fluorescence emerging at the 
+other end of the waveguide. Characteristics are found to be 
+drastically different between the gain materials as they are 
+confined to the liquid-air interface. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+(e) 
+ 
+(f) 
+Fig. 5: Ray-tracing simulation using FRED for two liquid waveguide structures looking from the top down. In both cases, core (liquid dye 
+solution) indicated with the lightly shaded region which is embedded in the darker cladding region. a. for multimode at liquid-air 
+interface with 110 micron width (Waveguide II); b. for multimode at liquid-PDMS interface (Waveguide II); c. for multimode at liquid-air 
+interface 14 micron width (Waveguide II) and d. for multimode at liquid-PDMS interface (Waveguide II) ; e. Mode field distribution in case 
+of liquid air interface for waveguide I; f. Mode field distribution in case of liquid air interface for waveguide II 
+ 
+ 
+
+(a)
+(b)(c)
+(d)(mt
+100
+Ax/s
+X Axis
+(uw)
+Local
+0.3
+3
+once
+0.2
+2
+n
+0
+0
+Local
+X Axis
+1
+Local(mm
+2
+0.02
+02
+Axis
+00
+(mm)
+0.020.00
+ueal
+12
+200
+100
+100
+0.00
+0.024
+Axis
+.00
+02
+Axis  
+ 
+ 
+Characterization 
+For any waveguide structure, there is a range of ray angle that will 
+fulfill the total internal reflection condition based on relative 
+refractive index difference between the core and clad region. In this 
+case, dye solutions with different concentration act as the core 
+medium and PDMS/air act as clad. The number of TIR for light is 
+inversely proportional to the diameter or cross-section of 
+microchannel. Ray tracing simulation platform (FRED) is used to 
+understand the propagation of fluorescence light 532 nm through 
+dye filled microstructure. Optical losses at the liquid-air interface 
+and liquid-PDMS interface in case of multimode and single mode 
+microstructure respectively is obtained as shown in Fig. 5. To 
+illustrate this, Fig. 5 shows a ray-trace simulation of a liquid core 
+waveguides. Gaussian beam from a coherent laser source is coupled 
+at the one end of waveguide with the help of 10X objective lens for 
+both structures. The laser light source is illuminated at the normal 
+incidence of the waveguide. Dye solution is filled inside the 
+microstructure. Above simulation has been applied by considering 
+the liquid dye with R.I. of 1.4030 as the core medium embedded 
+inside PDMS with R.I. of 1.40 and air with R.I. of 1 as the substrate. 
+Outside the core, lower clad being PDMS (1.40) and upper clad 
+being (Air =1), lower index region is formed. 
+The result obtained for different cases, shows that light can be 
+coupled inside the microstructure filled with 1mM concentrated 
+dye solution and confirms its waveguide nature. It also clears from 
+this study that optical losses at liquid-air interface is comparatively 
+less than that of liquid-PDMS interface irrespective of the 
+dimensions of waveguide. However, dimensions of waveguide 
+affect the total internal reflection per unit length. It is observed that 
+waveguide structure with smaller diameter is more suitable to act 
+as liquid mode guiding structure leading to increased probability of 
+guiding more number of photons to reach the output end. 
+These results confirm that laser light can propagate through 2D 
+liquid core waveguide structure by satisfying condition of total 
+internal reflection over the interface of liquid core and PDMS/Air 
+clad. By above observations, it becomes clear that many 
+complications and challenges can be easily overcome for 
+propagating index guided mode when air is used as a cladding 
+material. 
+In this experiment, we have filled the dye solution mix of ethanol 
+and benzyl alcohol into two microchannels (15 mm length each), 
+with 110 micron and 14 micron width respectively, on PDMS chip. 
+The end fire coupling method is used for optical characterization of 
+the developed liquid waveguide structures. The schematic of 
+characterization set up is as shown in Fig. 6 above. Here, the light 
+from Nd:YAG laser is end coupled into waveguide I and waveguide II 
+by using objective lens  and assembly of optics is also shown in Fig. 
+6. The roughness of PDMS wall for 2D microchannel for both 
+waveguide I and II were approximately limited to 1 micrometer due 
+to the better quality of direct writing of femtosecond laser. To 
+characterize the chip, we have used a micro syringe to insert the 
+liquid dyes into the microchannels as the core medium.  The 
+required liquid dyes for core medium are obtained by using ethanol 
++ benzyl alcohol as the host solution with two different solutes Rh-
+6G and Rh-101 to form two different dyes. Respective mixtures of 
+these two solutes in varying concentrations act as liquid cores 
+within the two microstructures. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 6: Characterization setup for liquid core /Air Cladding 
+waveguiding 
+ 
+As the absorption spectra of Rh-6G and Rh-101 lies in visible 
+wavelength therefore we have selected the Nd:YAG laser with 4 
+mW power and 7 nsec pulse duration with rep rate 10 Hz as the 
+pump source. This Nd: YAG laser is used to excite the fluorescent 
+dye molecules dissolved in the liquid core. The source is aligned to 
+beam iris and 10X objective lens. Across the objective lens beam 
+spot size is reduced to~100 µm for waveguide II structure and 10 
+micron for waveguide I structure. As the light and liquid are 
+pumped simultaneously to the microchannel, due to high refractive 
+index difference between liquid core and air, the fluorescence light 
+is guided and captured at the other end of microchannel. The outlet 
+end is connected to optical spectrometer. Fluorescence spectrums 
+are measured by changing the laser power and concentration of 
+dyes.   
+Model cross-sectional analysis for waveguide structures: In these 
+two structures as shown in Fig. 3 and 7, first one is multimode 
+waveguide II structure that allows multimodal tuning of waveguides 
+from liquid core and other one waveguide 1 support few modes 
+propagation.  
+To separate the fluorescence signal and excitation light, we need 
+‘Spectroscopic analysis and it is quite a difficult job to separate 
+these two outputs over the output end of channel. The intensity 
+profile for fluorescent light generated and propagated through the 
+developed liquid waveguide structures have been measured using 
+‘near-field intensity profile measurement’ experimental set up as 
+shown in Fig. 6 above.   
+ The output profiles for both waveguide structures have been 
+captured using CCD equipped with band-pass light filter for pump 
+light (λ=532 nm). Intensity at the output end of liquid waveguide 
+structure and corresponding intensity profile is shown in figure 7. 
+Profile measurements make it clear that the fabricated 
+microstructures are supporting the index guided modes for the 
+propagation and can be used as a waveguide like structure for 
+various applications. The small beam size (~100 µm) of the input 
+beam, relative to that of the liquid core (100 µm), helps in reducing 
+the coupling losses of pump light at the cross-section of the 
+microchannel.  Increment in the coupling and propagation losses 
+are due to the increasing effects of the scattering and diffraction of 
+the visible light through the PDMS channel walls (i.e., air/dye 
+solution/PDMS interfaces at the front and the end) with a normal 
+incident angle. 
+ 
+ 
+ 
+ 
+
+LASER
+M1
+M2
+10xObikns
+OSA
+10XObjlens
+BeamPellicke
+Inket
+Oullet
+PDMS 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 7:Intensity distribution for light propagating through: (a) Waveguide I and (b) multimode Waveguide II liquid core/air 
+clad cross section 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Fig. 8: Comparative studies of emission spectra for Waveguide I, Waveguide II structure and cuvette for (a)Rh-6G and (b) Rh-101 dye 
+solution 
+ 
+ 
+Results and discussion  
+In order to confirm waveguide nature of dye filled 2-D 
+microstructures, we have studied the fluorescence spectroscopy for 
+3mM concentration of dye (Rh-6G) as a liquid medium in three 
+different configurations i.e.) quartz Cuvette, b) waveguide II 
+structure and c) waveguide I structure.  
+The fluorescence emission spectra are collected for three different 
+structures in order to obtain the effect of microstructure 
+dimensions on the emission output. It is observed that emission 
+spectral 
+peak 
+wavelength 
+is 
+changed 
+by 
+15 
+nm 
+from 
+microstructures to cuvette filled with same dye solution Rh-101 and 
+pumped to a uniform Nd: YAG laser at 4 mW power as shown in Fig. 
+8.b. Similar shift has been observed in case of Rh-6G which shown 
+in Fig. 8.a. Increase in output photon density confirms the coupling 
+of FL inside the waveguide structure. It is also clear from the above 
+fig that FWHM of FL spectra gets narrower from Cuvette to 
+waveguide structure I. The spectral narrowing effect is observed 
+due to the Fabry-Perot resonator formed by dye solution filled  
+ 
+liquid waveguide and solvent-air interfaces. This result confirms 
+that fluorescence light generated by dye solutions gets coupled 
+through microchannel and forms Fabry-Perot type oscillations 
+which lead us to the conclusion that 2D structure fabricated on the 
+surface of PDMS functions as a liquid core/air cladding waveguide 
+structure. In addition, consideration of these two waveguides and 
+quartz cuvette confirms that dynamics of fluorescence spectra also 
+changes. The intensity, lasing peak and line width change according 
+to dimensions of individual structure. Same results are observed for 
+Rh-101 dye solution. FWHM of fluorescence signal of quartz 
+Cuvette is observed 48.8 nm and peak wavelength at 637.59 nm. 
+In multimode waveguide II for Rh-101 dye, line width achieved is 
+13.53 nm, peak wavelength is 624.10 nm and that for waveguide I 
+structure line width is 6.94 nm and peak wavelength is 623.75nm. 
+In case of Rh-6Gdye solution, FWHM for Cuvette is 42.89 nm and 
+peak wavelength is 580.90 nm. For multimode waveguide II 
+structure is 14.52 nm, peak wavelength is 573 nm and for 
+  
+ 
+ 
+ 
+ 
+ 
+Waveguide 
+ 
+Waveguide II 
+(a) 
+(b) 
+Fluorescence 
+emission on 
+cross section 
+Pump 
+Pump 
+Fluorescence emission on 
+cross section 
+ 
+Rh-6G 
+(a) 
+Cuvette 
+Waveguide II 
+Waveguide I 
+ 
+Rh-101 
+(b) 
+           Cuvette   
+               Waveguide II 
+               Waveguide I 
+ 
+ 
+
+1.0
+0.9 
+Intensity
+0.8
+Normalized
+0.7
+0.6
+0.5
+0.4
+0
+5
+10
+15
+20
+25
+30
+35
+WaveguideDepth (um)70000
+Cuvette
+60000
+Multimode
+.U.
+Single mode
+A
+50000
+Count
+40000
+30000
+20000
+Ph
+10000
+0
+540
+560
+580
+600
+620
+640
+Wavelength(nm)70000
+Cuvette
+60000
+Multimode
+3
+Singlemode
+A
+50000
+Count
+40000
+hoton
+30000
+20000
+P
+10000
+0
+580600620640660680700720740
+Wavelength(nm)Intensity
+0.8
+Normalized
+0.6
+0.4
+0.2 
+0.0
+0
+20
+40
+60
+80
+100
+120
+WaveguideDepth (um)  
+ 
+ 
+waveguide I structure linewidth reduces to 5.34 nm, peak 
+wavelength is shifted at 573.70 nm. Through a comparison study, it 
+has been observed that peak wavelength in multimode waveguide I 
+and II structure is quite less (blue shifted) compare to Cuvette 
+output. In the present study, we can see for quartz Cuvette the 
+output florescence spectrum has a large bandwidth. Due to the 
+small dimensions of microchannels, the obtained graph clearly 
+indicates that the linewidth of waveguide structure II is less than 
+that of Cuvette and waveguide structure I have even lower line 
+width compared to the structure II. 
+Effect of power for higher concentration regime 
+For characterization of these FS written microchannels in terms of 
+multimode waveguide microstructures I and II, we have studied the 
+effect of pump power for Rh-6G and Rh-101 dye solutions. It has 
+been observed that with variation in the pump power, a significant 
+tunability has been observed in fluorescence spectra. All these 
+measurements have been observed at the room temperature. Fig.9 
+illustrates the measured emission spectra with Rh-6G for 10 mM in 
+both liquid core/air waveguide structure I and II. Here we have 
+varied the input power in a range of 4 - 12 mW for both cases and 
+observed that for lower concentrations, insignificant change was 
+observed in the fluorescence peak wavelength in correspondence 
+to the variation in incident laser power but for 10 mM, peak 
+wavelength shift has been observed as power is varied. A 
+florescence peak wavelength count emerges as optical pumping 
+power density is increased. The absorption of incident laser beam is 
+responsible for change in refractive index gradient of dye solution 
+of the order of 10-3 to 10-4 due to optically heated thermal lensing 
+effect [11].  Also, incident pulsed high-power laser beam generates 
+acoustic pressure waves inside the dye filled liquid waveguide 
+structure which induce the variation in the refractive index of 
+medium [11, 12].   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+In this way, incident laser power plays significant role in the shift of 
+florescent peak wavelength and output spectrum which is reflected 
+in the experimental results as shown in the Fig. 9 respectively. In 
+low concentration regime, isolated dye molecules are present but 
+as we increase the concentration of dye, the spacing between dye 
+molecules decreases and aggregates are formed.  
+Thus, peak wavelength variation can be seen in very high 
+concentration regime. The other phenomenon which contributes to 
+the modified output spectra of dye is ‘self-absorption’ due to higher 
+concentrations. As the molecular dimmer are formed at high 
+concentration, it explains the appearance of a second shift in 
+measured fluorescence spectroscopy such that red shift is observed 
+for 10 mM dye concentration by varying the power from 4 mW to 
+12 mW. From Fig. 9, we can clearly observe the peak wavelength 
+for multimode waveguide structure II for Rh6G   solution was 
+achieved at 579.8 nm at 4mW pump power. As the power increases 
+to 6mW, peak wavelength is shifted at 581.42 nm. By varying to 
+higher power, red shifted peak wavelength is reached up to 583.25 
+nm. Same experiment has been repeated for Rh-101 dye solution. 
+We took 10 mM solution and measured the fluorescence spectra 
+for multimode waveguide structure II at 2 mW, the peak 
+wavelength is captured at 626.48 nm. The amount of light guided 
+inside the multimode mode waveguides I and II are strongly 
+dependent on the refractive index difference between ncore and nclad 
+as: 
+∆������������ = ������������������������������������������������ − ������������������������������������������������ 
+The Rh-6G and Rh-101 are dissolved in mixture of ethanol +benzyl 
+alcohol as a host solution.   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+Rh-6G Waveguide I 
+ 
+ Rh-6G Waveguide II 
+ 
+ Rh-101 Waveguide I 
+ 
+Rh-101 Waveguide II 
+(c) 
+(d) 
+(b) 
+(a) 
+Fig.  9: Study of pump power-based fluorescence emission spectra for(a) Rh6G in waveguide structure I (b) Rh6G in 
+multimode waveguide structure II (c) Rh101 in waveguide structure I and (d) Rh101 in waveguide structure II 
+(A=4mW, B=6 mW, C=8mW, D=10mW E=12mw) 
+ 
+ 
+
+70000
+A
+B
+60000
+c
+3
+3
+50000
+E
+Counts
+40000
+Photon
+30000
+20000
+10000
+0
+560
+570
+580
+590
+600
+610
+620
+630
+Wavelength (nm)70000
+60000
+(A.U.)
+50000
+Counts
+40000
+Photon
+30000
+20000
+10000
+0 -
+580
+600
+620
+640
+660
+680
+Wavelength (nm)70000
+60000
+(A.U.)
+50000
+Counts
+40000
+Photon
+30000
+20000
+10000
+0
+550
+560
+570
+580
+590
+600
+610
+Wavelength (nm)70000
+60000
+(wu)
+50000
+Counts
+40000
+Photon
+30000
+20000
+10000
+0
+580
+600
+620
+640
+660
+680
+Wavelength (nm) 
+ 
+ 
+ 
+The experiment is repeated for waveguide structure I for same 
+solution. Light is coupled from the output end of waveguide. Light 
+after being guided inside the waveguide structure I is observed at 
+the cross section of the waveguide and we can observe in the graph 
+that the peak wavelength and line width for same power and 
+concentration changes. There is slight change in Peak wavelength 
+but line width drastically changes in the waveguide I compared to 
+the waveguide II.  
+For waveguide structure I, the red shift in to fluorescence emission 
+peak for both dyes caused by the variation pump power from 4 mW 
+to 12 mW with step size of 2 mW. The corresponding tunability 
+achieved is in the range of 579.87-583.25 nm and average line 
+width is 6.8 nm in case of Rh-6G. For the waveguide structure I, in 
+case of Rh-101 based active solution, tunability achieved is 7 nm 
+and observed average line width is 6 nm. For Rh-101 dye in 
+waveguide structure I tunability achieved is in the range of 620.49- 
+628.44 nm. In the case of waveguide structure II, for Rh-6G, red 
+shift in peak wavelength has been observed. Tunability of peak 
+wavelength being 4 nm and average line width being 10 nm. In the 
+case of Rh-101, the tunability of 6 nm is achieved. For multimode 
+waveguide structure II in case of Rh-101, spectral tunability is 
+achieved in the range of 626.48 - 632.50 nm and average line width 
+is 10 nm. In case of Rh-6G, for same multimode waveguide II, 
+spectral tunability is achieved in the range of 579.87-583.25 nm and 
+average FWHM line width is 9.5 nm. 
+ 
+Effect of concentration  
+The tunability in output band of liquid filled microstructures is 
+mainly determined by selection of dye solution and its solubility 
+limit to highly dilute systems of Rh-6G and Rh-101. In case of lower 
+concentration regime (concentrations of 0.1 mM), component of 
+self-absorption is quite significant which decrease the intensity of 
+signal. In addition, at higher concentrations regime (concentrations 
+of 10 mM), intermolecular self-quenching rapidly decreased the 
+output intensity [11]. Particularly, in high concentration regime, the 
+Rh-6G and Rh-101 molecules arrange themselves into H type and J 
+type dimmers [14, 15 & 16]. This dimmer formation changes the 
+electronic structure and as a result, the output emission spectrum is 
+also changed. In this way, the variation in the concentration of 
+liquid medium provides an optical flexibility for liquid waveguide 
+structures.  
+The experimental observation for spectral dependency of liquid 
+waveguide structures for varying concentration of Rh-6G and Rh-
+101 dye solution at fixed pump power is as shown in below Fig. 
+(10). The detailed analysis of output spectra for Rh-6G 
+concentrations ranging from 1 mM to 4 mM and Rh-101 
+concentrations ranging from 1 mM to 5 mM have been done. 
+  
+It was observed that the spectral position of the propagating mode 
+through the liquid waveguide structure shifts toward longer 
+wavelengths by increasing the concentration of dye solution. In 
+case of waveguide I filled Rh-6G solution, the peak wavelength shift 
+observed from 573.16 nm to 580.67nm for 1mM to 4 mM 
+concentration change. Along with peak wavelength, average line 
+width shift is also observed from to 5 -6.01 nm for the same. For Rh-
+101 filled waveguide I , 5nm shift in peak wavelength and ± 2 nm 
+sift in line width is observed when concentration changes from 1 
+mM to 5 mM respectively. As Fig. 10 shows, the wavelength of the 
+peak maximum is red-shifted with varying concentration. The same 
+experiments were carried out for multimode waveguide structure II 
+for both dye solutions. Similarly, spectral study for different 
+concentrations in multimode structure II for Rh-6G dye, 8 nm red 
+shift in peak wavelength and 1.5 nm shift in linewidth have been 
+observed while 5 nm peak wavelength red shift with ± 2 nm 
+linewidth shift has been observed for Rh-101 respectively. Here, the 
+peaks occurred at different wavelengths as per the changing 
+concentration of liquid medium. Red shift in the output spectra is 
+observed when concentration is increased from 1 mM to 4 mM. The 
+apparent red shift in the emitted intensity signal is due to the small 
+Stokes shift of Rh-6G and the large spectral overlap in absorption 
+and emission [13, 14]. Same observations have been seen for Rh-
+101 solution. The optimum optical absorption of pump beam, inside 
+the dye solution filled microchannel is achieved at concentration of 
+1 mM. 
+ 
+Photo bleaching effect in microstructure 
+ The rate of photo bleaching primarily depends upon the type of 
+dye, host material and their optical properties. Additionally, 
+illumined intensity of source, wavelength of source, exposure time 
+and temperature also affect the extent of photo bleaching [16, 17]. 
+Photo bleaching is not a desirable phenomenon for lab-on chip 
+based optofluidic waveguides and optofluidic lasers. It destructs the 
+continuous output of miniaturized device and limits its usage to 
+short time periods only. Here, we have studied the photo bleaching 
+effect in waveguide structure I and II for both Rh-6G and Rh-101 
+dye mediums. This study helps us to design and improve upon the 
+functionalities of optofluidic chips.   
+As a consequence of photo bleaching due to the long exposure of 
+pump intensity to the liquid active medium, the fluorophores lose 
+the ability to emit fluorescence in the same magnitude of intensity. 
+The linewidth and intensity of florescence output have been 
+significantly changed due to photo bleaching effect in the liquid 
+waveguide.  Due to diffusion dynamics in the presence of on chip 
+reservoirs, in case of micro dye lasers, the supply of unbleached dye 
+solution on faster time scale is not required. In the studied case, 
+length of microchannel is 15 mm and width is 110 micron (W/L= 
+0.0073) for waveguide structure II and for waveguide I (W/L= 
+0.00093). For both the cases longitudinal coupling of light have 
+been done in slit area.  Photo bleaching time for waveguides can be 
+converted to just a few minutes without using any costly liquid 
+handling devices and replacement. Here, we have used the static 
+phenomenon of liquid waveguides without using the external fluidic 
+handling systems such as syringe pumps. The experimentally 
+observed fluorescence dynamics is in qualitative agreement with 
+the bleaching-diffusion dynamics [17, 18 & 19]. 
+ 
+In Microsystems, photo bleaching creates unwanted intensity 
+changes in the output. The quantum yield of photo bleaching and 
+the molar extinction coefficient are inherent properties of 
+Rhodamine-6G and Rhodamine-101. In case of static measurements 
+inside microstructures, most affricating factors for photo bleaching 
+inside waveguide structures filled with diluted solutions can be 
+determined by applying Beer’s law as [14]: 
+ 
+������������������������������������������������ = ������������������������������������������������������������������������ (−������������������������������������������������������������������������������������������������������������������������������������) 
+ 
+Aout is the amount of emitting molecules remaining after photo 
+bleaching; Ain is the original concentration of absorbed dye 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ Rh-6G waveguide I 
+ 
+Rh101 waveguide I 
+ 
+Rh101waveguide II 
+ 
+Rh-6G (waveguide I) 
+(b) 
+ 
+(c) 
+Rh-101 (waveguide II) 
+ 
+Rh-101 (waveguide I) 
+(d) 
+ 
+(a) 
+Rh-6G (waveguide II) 
+(c) 
+(d) 
+ 
+Rh-6G waveguide II 
+(b) 
+(a) 
+Fig. 10: Studies of concentration variation-based fluorescence emission spectra for multimode waveguide structures I and II 
+for Rh6G and Rh101: (a) Rh6G filled structure I (b) Rh6G filled multimode structure II (c) Rh101 filled structure I (d) Rh101 
+filled multimode structure II. 
+Fig. 11: Photobleaching studies for (a) Rh6G in multimode structure II (b) Rh6G in structure I (c) Rh101 in multimode structure II 
+and (d) Rh101 in waveguide structure I 
+ 
+ 
+
+70000
+1 mM
+(n'v)
+60000
+2 mM
+3 mM
+50000
+4mM
+Counts
+40000
+30000
+Photon
+20000
+10000
+0
+560
+580
+600
+620
+640
+Wavelength (nm)70000
+60000
+o S
+A.U.
+10 S
+50000
+20 S
+Count
+30 S
+40 S
+40000
+60 S
+70 S
+Photon
+30000
+20000
+10000
+0
+550
+560
+570
+580
+590
+600
+610
+Wavelength (nm)70000
+(A.U.)
+oS
+60000
+25 S
+50 S
+50000
+75 S
+Counts
+100 S
+40000
+125 S
+150 S
+hoton
+175 S
+30000
+200 S
+225 S
+P
+20000
+10000
+580
+600
+620
+640
+660
+680
+700
+Wavelength (nm)70000
+10 S
+60000
+20 S
+30 S
+(A.)
+50000
+40 S
+Counts
+50 S
+60 S
+40000
+70 S
+80 S
+hoton
+30000
+90 S
+20000
+10000
+0
+580
+600
+620
+640
+660
+680
+Wavelength (nm)70000
+os
+60000
+20 S
+(A.U.)
+40 S
+50000
+60 S
+Count
+80 S
+40000
+100 S
+120 S
+Photon
+30000
+140 S
+160 S
+20000
+180 S
+10000
+0
+560
+580
+600
+620
+Wavelength(nm)70000
+1mM
+60000
+2 mM
+3 mM
+50000
+4mM
+Counts
+40000
+Photon
+30000
+20000
+10000
+0
+560
+580
+600
+620
+640
+Wavelength (nm)70000
+1 mM
+60000
+2 mM
+(A.U)
+3 mM
+50000
+4mM
+Count
+5 mM
+40000
+30000
+20000
+10000
+0
+600
+610
+620
+630
+640
+650
+660
+Wavelength (nm)70000
+1 mM
+2 mM
+(A.U
+60000
+3 mM
+4 mM
+Counts
+50000
+5 mM
+40000
+noton
+30000
+20000
+10000
+0
+600
+610
+620
+630
+640
+650
+Wavelength(nm) 
+ 
+ 
+ 
+molecules, I0 is the incident light irradiance, Qph is the quantum 
+yield of photo bleaching and te is the exposure time. From the 
+above equation, it is clear that quantity of photo bleached 
+molecules inside the solution is exponentially dependent on 
+exposure time and pump intensity. Therefore, even a small increase 
+in time or light intensity results in a substantial increase in the 
+amount of photo bleaching. Our experimental results reveal that 
+these optofluidic waveguides can be operated over a few minutes 
+without needing a flow of fresh dye solution as shown in Fig. (11). In 
+case of Rh-6G solution, photo bleaching time is observed to be 70 
+sec for waveguide I and 180 sec for multimode waveguide structure 
+II, while in case of Rh-101, photo bleaching time is observed to be 
+90 sec and 225 sec for multimode waveguide structure I and II 
+respectively. 
+This experiment confirms that decay time of Rh-101 is slightly 
+greater than that of Rh-6G. This observed behavior is justified by 
+previous publication [17, 20]. The photo bleaching time can further 
+be improved by a factor of 3 to 4 times by adding reservoirs on chip. 
+Also, by converting the fabricated 2D structures into a 3D chip and 
+using different pumping scheme, the developed liquid waveguide 
+structure can be used in established optofluidic devices with 
+enough output which would be sufficient and even more than is 
+required to do the lab-on-chip experiments 
+ 
+Conclusion:  
+In conclusion, we have demonstrated a novel femtosecond 
+fabricated liquid-core/air-clad waveguide microstructures on a 
+PDMS microchip. We have studied the role of concentration, photo-
+bleaching and incident power on the output of waveguides in detail. 
+This work gives a very good understanding towards the interaction 
+of light and fluid in micro dimension. Tunability in the form of 
+intensity, wavelength and linewidth has been successfully obtained. 
+The characteristic of these waveguide sources can be easily 
+controlled and modulated by adjusting the fluid properties of the 
+core medium. After converting these 2D chips into 3D chips and 
+adding some optical component to the same, the liquid waveguide 
+source can be made into a tunable optofluidic laser having a 
+coherent light source that can be integrated with multifunctional 
+lab-on chip systems. In this way, fluorescence measurement and 
+detection by optofluidic devices can provide a powerful platform 
+for analysis of biological systems and aid significantly in medical 
+diagnostics and chemical detection. This research gives a brief idea 
+about development and maintenance of highly functional lab-on-
+chip waveguides which can be used out of laboratory also for many 
+applications. 
+ 
+Acknowledgement: 
+We acknowledge the support provided by CMTI Bangalore, India for 
+femtosecond micromachining fabrication facility.       
+ 
+Reference:  
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+12. Shane M. Eaton,Carmela De Marco,Rebeca Martinez-
+Vazquez,Roberta 
+Ramponi,Stefano 
+Turri,Giulio 
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+“Femtosecond 
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+microstructuring for polymeric lab-on-chips” Journal of 
+Biophotonics, 2012, 5(8-9). 
+13. Penzkofer. W. I.eupacher et al., “ Fluorescence  behaviour 
+of highly concentrate rhodamine 6G solutions”. journal of 
+Luminescence, 1987,37, 61-72. 
+14. Florian M. Zehentbauer et al., “Fluorescence spectroscopy 
+of Rhodamine 6G: Concentration and solvent effects”,  
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+SpectrOscopy, 2014, 121() 147-151.                                                                                       
+15. K. Noack, J. Kiefer, A.I.saipem, et al., “Concentration 
+dependent hydrogen bonding effects on the dimethyl 
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+methanol and ethano”l, ChemPhysChem 2010, 11, 630-
+637. 
+16. VJ. Gavrilenko, MA  Noginov, et al., “Ab  initio study of  
+optical  properties  of Rhodamine 6G molecular dimers”, 
+Journal of Chemical Physic, 2006s 124, 044301. 
+17. Morten Gersborg-Hansen et al., “Bleaching and diffusion 
+dynamics in optofluidic dye lasers”, APPLIED PHYSICS 
+LETTERS, 2007,90, 143501. 
+18. Jerker Widengreny et al.,   “Mechanisms of photobleacing 
+investigated by fluorescence correlation spectroscopy”, 
+Bioimaging, 1996, 4, 149–15. 
+19. Mingyu Chapma et. al., “Rhodamine 6G Structural 
+Changes in Water/Ethanol Mixed Solvent”, Journal of 
+Fluorescence, 2018, 28:1431–1437. 
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+the Fluorescence Quantum Yield of Semiconductor 
+Nanocrystals”, Materials, 2011, 4, 1182-1193. 
+ 
+ 
+

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+arXiv:2301.02921v1  [math.AP]  7 Jan 2023
+Non-local optimized Schwarz method
+with physical boundaries
+X.Claeys1
+1Sorbonne Université, Laboratoire Jacques-Louis Lions
+Abstract
+We extend the theoretical framework of non-local optimized Schwarz methods as in-
+troduced in [Claeys,2021], considering an Helmholtz equation posed in a bounded cavity
+supplemented with a variety of conditions modeling material boundaries. The problem is
+reformulated equivalently as an equation posed on the skeleton of a non-overlapping parti-
+tion of the computational domain, involving an operator of the form "identity + contrac-
+tion". The analysis covers the possibility of resonance phenomena where the Helmholtz
+problem is not uniquely solvable. In case of unique solvability, the skeleton formulation
+is proved coercive, and an explicit bound for the coercivity constant is provided in terms
+of the inf-sup constant of the primary Helmholtz boundary value problem.
+Introduction
+Large scale simulation of harmonic wave propagation phenomena remains a challenge in the
+context of which one of the most effective substructuring domain decomposition methods
+(DDM) was introduced by Després [10]. Commonly referred to as Optimized Schwarz Method
+(OSM), it consists in local solves of the wave equation, maintaining a coupling between sub-
+domains through a reformulation of transmission conditions in terms of ingoing and outgoing
+Robin traces. The new transmission conditions involve an exchange operator that swaps traces
+from both sides of each interface between neighboring subdomains. This approach was put
+in a general theoretical framework in [9] and we point to [14] for an overview of this type of
+strategy.
+In a discrete setting, the appropriate definition of the exchange operator raises issues at
+cross-points, where at least three degrees of freedom have to communicate, because it is then
+unclear what should be the discrete counterpart of swapping. Although several heuristics had
+been proposed in the literature for dealing with this situation [12, 13, 19, 11, 1], most strategies
+based on this local swapping operator experienced deteriorated performance in the presence
+of cross points.
+In a series of articles [5, 6, 7, 8], we proposed a variant of OSM where the usual local swap-
+ping exchange operator is replaced by an alternative a priori non-local operator that naturally
+accommodates the presence of cross-points. This new approach can cope with arbitrary sub-
+domain partitions, with a possibly very complicated wire basket. In [5], we analyzed this new
+approach at the continuous level considering a transmission problem posed on the full space
+1
+
+Rd, and the formulation associated to this new DDM strategy was proved strongly coercive,
+which paved the way to convergence estimates for linear solvers (e.g. Richardson, GMRes).
+This novel approach was adapted to a finite element discretised setting and a full conver-
+gence theory was developed in [8, 6]. In passing, this new theoretical framework covered the
+case of the original Després algorithm hence offering a genuine generalization. The whole the-
+ory was confirmed by numerical results both in 2D and 3D. While the previous developments
+were concerned with scalar harmonic wave propagation, the case of Maxwell’s equations was
+considered in [7, 20].
+In the present contribution we extend the theory of [5] in several directions. First of all, while
+[5] considered only the case of a transmission problem posed on the whole of Rd, we consider
+here the case of a cavity problem posed in a bounded domain Ω ⊂ Rd. This boundary value
+problem takes the form
+div(µ−1∇u) + κ2u = −f in Ω
++ boundary condition on ∂Ω.
+(1)
+Here again we reformulate it as an equation in terms of traces posed on the skeleton of the
+subdomain partition, which we call skeleton formulation. While in previous contributions the
+problem had been assumed uniquely solvable (see e.g. [8, §1] or [6, §1.2]), the analysis is
+here extended so as to cover the case where (1) is not necessarily uniquely solvable which
+covers the case of non-trivial resonance phenomenon. The skeleton formulation is then proved
+uniquely solvable if and only if this holds for (1) and, if this condition is fulfilled, the skeleton
+formulation is proved to be strongly coercive. Although coercivity was already established
+in [5], we provide in addition an explicit estimate of the coercivity constant in terms of the
+inf-sup condition of the primary variational formulation.
+Our whole analysis rests on an interpretation of the properties of (1) in terms of a pair of
+two closed linear manifolds: one that models transmission conditions, and another one that
+models local wave equations. Studying properties of operators by means of pairs of closed
+linear manifolds follows the spirit of [16, iv.4 & iv.5].
+Like [5], the present contribution is purely theoretical. It aims at laying solid analytical
+foundations for a better understanding of the spectral properties of the skeleton formulation,
+which is important in the perspective of devising both computationally efficient eigensolvers
+and domain decomposition preconditionners. We do not provide any numerical experiment.
+Such results shall be presented in a forthcoming contribution that will develop a discrete
+variant of the present analysis, in the spirit of [8, 6].
+The outline of this article is as follows. In the first two sections we introduce general notations
+for both Hilbert analysis and Sobolev spaces, including trace operators, Dirichlet-to-Neumann
+maps and harmonic liftings. Next we describe the problem under study, specifying precisely
+the assumptions underlying our analysis, which allows in particular to deal with a variety
+of boundary conditions. How to apply this framework for common boundary conditions is
+illustrated with examples. Further notations are introduced for dealing with multi-domain
+configurations. This leads in particular to a characterization of transmission conditions based
+on a non-local exchange operator, see Proposition 4.3, which had been an important innovation
+of [5]. We use this multi-domain formalism to re-express the boundary value problem under
+study. The kernel and the range of this operator are then re-interpreted in terms of a pair of
+closed linear manifolds. One manifold models wave equations local to each subdomain, and
+2
+
+the other one models transmission conditions. Wave equations local to each subdomain are
+then re-expressed by means of a so-called scattering operator, which we use to finally provide a
+formulation involving tuples of Robin traces on the skeleton of the subdomain partition. This
+skeleton formulation is proved to systematically admit closed range, and its kernel is put in
+correspondence with the kernel of the original formulation. Finally we prove strong coercivity
+for the skeleton formulation and derive an estimate for the coercivity constant that is explicit
+with respect to the inf-sup constant of the original variational formulation.
+1
+General notation conventions
+We first set a few general notation conventions regarding analysis in Banach spaces. All vector
+spaces that we are going to consider have C as scalar field. Assuming that H is a Banach
+space equipped with the norm ∥ · ∥H, its topological dual denoted H∗ will systematically be
+equipped with the norm
+∥ϕ∥H∗ =
+sup
+v∈H\{0}
+|ϕ(v)|
+∥v∥H
+.
+(2)
+The canonical duality pairing will be systematically denoted ⟨·, ·⟩ : H∗×H → C and defined by
+⟨ϕ, v⟩ := ϕ(v). Although the space H does not appear explicitly in the notation "⟨ϕ, v⟩", when
+such pairing angle brackets are used, it shall be clear from the context which pair of spaces
+(H, H∗) is under consideration. We emphasize that the duality pairings we consider do not
+involve any complex conjugation. We shall write ⟨v, ϕ⟩ = ⟨ϕ, v⟩ ∀v ∈ H, ϕ ∈ H∗ indifferently.
+For any subset X ⊂ H, we denote its polar set by
+X◦ := {ϕ ∈ H∗, ⟨ϕ, v⟩ = 0 ∀v ∈ X}.
+(3)
+Assuming that V is another Banach space equipped with the norm ∥ · ∥V, and L : H → V is a
+bounded linear map, we shall refer to its inf-sup constant denoted and defined as follows
+infsup
+H→V
+(L) :=
+inf
+u∈H\{0}
+∥L(u)∥V
+∥u∥H
+(4)
+In the case where L is invertible, this inf-sup constant equals the inverse to the continuity
+modulus of L−1. The inf-sup constant is well defined even if L is not invertible though. The
+adjoint to the map L : H → V shall be defined as the unique bounded linear map L∗ : V∗ → H∗
+satisfying
+⟨L∗(p), u⟩ := ⟨p, L(u)⟩
+(5)
+for all p ∈ V∗ and all u ∈ H. Once again, we insist that no complex conjugation comes into
+play in (5). The bounded linear map L induces another bounded linear map L : H → V
+defined by L(u) := L(u) for all u ∈ H.
+A bounded linear operator T : H → H∗ is called self-adjoint if T = T∗ and, in this case we
+have ⟨T(u), u⟩ ∈ R for all u ∈ H. It is called positive definite if ⟨T(u), u⟩ ∈ (0, +∞) for all
+u ∈ H\{0}. If T is both self-adjoint and positive definite, the sesquilinear form u, v �→ ⟨T(u), v⟩
+induces a scalar product over H and the associated norm is denoted
+∥u∥T :=
+�
+⟨T(u), u⟩.
+(6)
+3
+
+We shall also consider cartesian products H1 × · · · × HJ where each Hj is a Banach space
+equipped with the norm ∥ · ∥Hj.
+Then the cartesian product shall be equipped with the
+following canonical norm and duality pairings
+∥v∥2
+H1×···×HJ := ∥v1∥2
+H1 + · · · + ∥vJ∥2
+HJ
+⟨v, q⟩ := ⟨v1, q1⟩ + · · · + ⟨vJ, qJ⟩.
+(7)
+for v = (v1, . . . , vJ), vj ∈ Hj, and q = (q1, . . . , qJ), qj ∈ H∗
+j. If Vj, j = 1, . . . , J is another
+collection of Banach spaces and Lj : Hj → Vj are bounded linear maps, we shall also consider
+the block-diagonal operator diag(L1, . . . , LJ), mapping H1 × · · · × HJ into V1 × · · · × VJ and
+defined, for v = (v1, . . . , vJ), and q = (q1, . . . , qJ), by
+⟨q, diag(L1, . . . , LJ) v⟩ := ⟨q1, L1(v1)⟩ + · · · + ⟨qJ, LJ(vJ)⟩.
+2
+Single domain functional setting
+Now we need to introduce classical function spaces.
+For any Lipschitz open set ω ⊂ Rd,
+we consider L2(ω) := {v : ω → C measurable, ∥v∥2
+L2(ω) :=
+�
+ω |v(x)|2dx < +∞} and define
+Sobolev spaces
+H1(ω) := {v ∈ L2(ω), ∇v ∈ L2(ω)d}
+∥v∥2
+H1(ω) := ∥∇v∥2
+L2(ω) + γ−2∥v∥2
+L2(ω)
+(8)
+where γ > 0 is a real positive parameter. Incorporating γ-dependency in the norm will allow
+to establish γ-uniform estimates in the sequel. The space H1
+0(ω) will refer to the closure of
+D(ω) := {ϕ ∈ C ∞(Rd), supp(ϕ) ⊂ ω, supp(ϕ) bounded} for ∥ · ∥H1(ω).
+Next we introduce the space of Dirichlet traces H1/2(∂ω) := {v|∂ω, v ∈ H1(Rd)} equipped with
+the quotient norm ∥v∥H1/2(∂ω) := min{∥ϕ∥H1(Rd), ϕ ∈ H1(Rd) and ϕ|∂ω = v}. The topological
+dual to H1/2(∂ω) will be denoted H−1/2(∂ω) = H1/2(∂ω)∗. As detailed for example in [17,
+Thm.3.38], the trace map gives rise to a bounded linear operator
+Bω : H1(ω) → H1/2(∂ω)
+Bω(v) := v|∂ω
+∀v ∈ D(Rd).
+(9)
+We underline that Bω refers to the trace taken from the interior of ω. The norm (8) gives rise
+to a natural right-inverse of this Dirichlet boundary trace operator. We define the harmonic
+lifting operator B†
+ω : H1/2(∂ω) → H1(ω), see [21, §1.2.2.4], through norm minimization
+Bω · B†
+ω(v) = v
+∀v ∈ H1/2(∂ω) and
+∥B†
+ω(v)∥H1(ω) := min{∥φ∥H1(ω), Bω(φ) = v, φ ∈ H1(ω)}.
+(10)
+Denote H1(∆, ω) := {v ∈ H1(Ω), ∆v ∈ L2(Ω)} and let nω refer to the unit normal vector
+field to the boundary ∂ω directed toward the exterior of ω.
+The Dirichlet trace operator
+ϕ �→ ϕ|∂ω, resp. the Neumann trace operator ϕ �→ nω · ∇ϕ|∂ω, can be extended by density as
+a bounded linear map H1(ω) → H1/2(∂ω) resp. H1(∆, ω) → H−1/2(∂ω), see e.g. [17, Lem.4.3].
+4
+
+The Dirichlet-to-Neumann (DtN) map Tω : H1/2(∂ω) → H−1/2(∂ω) is defined as the unique
+bounded linear operator satisfying
+Tω(φ|∂ω) := nω · ∇φ|∂ω
+∀φ ∈ H1(∆, ω) satisfying
+− ∆φ + γ−2φ = 0
+in ω.
+(11)
+This is a real valued and self-adjoint operator Tω = Tω and T∗
+ω = Tω which induces a scalar
+product over H+1/2(∂ω) and the Neumann-to-Dirichlet map T−1
+ω
+: H−1/2(∂ω) → H+1/2(∂ω)
+induces a scalar product over H−1/2(∂ω). We set
+∥v∥2
+Tω := ⟨Tω(v), v⟩
+∥p∥2
+T−1
+ω
+:= ⟨T−1
+ω (p), p⟩.
+(12)
+It is a well established fact (see e.g.
+[21, Def.1.41] or [23, §6.6.3]) that ∥ · ∥H1/2(∂ω) and
+∥·∥H−1/2(∂ω) are equivalent to the norms (12). Applying the Euler equation characterizing the
+harmonic lifting B†
+ω(v) as unique solution to the minimization (10), see e.g. [4, Thm.7.2-1],
+we have −∆B†
+ω(v) + γ−2B†
+ω(v) = 0 in ω, so that Tω(v) = nω · ∇B†(v)|∂ω. We also deduce
+that ∥φ|∂ω∥Tω = ∥B†
+ω(φ|∂ω)∥H1(ω) ≤ ∥φ∥H1(ω) for all φ ∈ H1(ω) and, in particular, we have
+the inequalities
+∥B†
+ω(v)∥H1(ω) = ∥v∥Tω
+∀v ∈ H1/2(∂ω),
+∥Bω(u)∥Tω ≤ ∥u∥H1(ω)
+∀u ∈ H1(ω).
+(13)
+3
+Single domain variational formulation
+The next step in our analysis will consist in writing Problem (1) in a variational form able to
+cope with a variety of boundary conditions. This is why we treat the boundary condition by
+means of an additional Lagrange parameter. Let Ω ⊂ Rd, Γ := ∂Ω refer to an open bounded
+Lipschitz set and its boundary and denote
+H(Ω × Γ) := H1(Ω) × H−1/2(Γ)
+Our analysis will start from a variational formulation of (1), later referred to as the primary
+formulation, that we write: find u ∈ H(Ω × Γ) such that
+AΩ×Γ(u) = ℓΩ×Γ
+(14)
+where the bilinear map underlying the variational problem is written as a bounded linear
+operator AΩ×Γ : H(Ω × Γ) → H(Ω × Γ)∗ assumed to systematically take the following form:
+for any u, v ∈ H1(Ω) and p, q ∈ H−1/2(Γ),
+Assumption:
+⟨AΩ×Γ(u, p), (v, q)⟩ := ⟨AΩ(u), v⟩ + ⟨AΓ(u|Γ, p), (v|Γ, q)⟩
+(A1)
+The map AΩ×Γ involves a volume part AΩ : H1(Ω) → H1(Ω)∗ that accounts for the Helmholtz
+equation in the interior of the domain Ω. For µ ∈ C and κ : Ω → C an essentially bounded
+5
+
+measurable function, it is assumed of the following form
+Assumptions:
+⟨AΩ(u), v⟩ :=
+�
+Ω µ−1∇u · ∇v − κ2uv dx,
+with ℑm{κ(x)2} ≥ 0, ∀x ∈ Ω
+supx∈Ω|κ(x)| < ∞
+ℜe{µ} > 0, ℑm{µ} ≥ 0.
+(A2)
+The assumptions above imply in particular that ℑm{⟨AΩ(u), u⟩} ≤ 0 ∀u ∈ H1(Ω).
+The
+operator AΩ×Γ also involves a pure boundary part AΓ that models boundary conditions,
+AΓ : Hb(Γ) → Hb(Γ)∗
+where Hb(Γ) := H1/2(Γ) × H−1/2(Γ).
+(15)
+The boundary operator AΓ involves traces on Γ and is chosen in accordance with the boundary
+conditions of our primary boundary value problem (1). We will need to rely on the following
+additional assumptions
+Assumptions:
+i) ℑm{⟨AΓ(u), u⟩} ≤ 0
+∀u ∈ Hb(Γ)
+ii) range(AΩ×Γ) is closed in H(Ω × Γ)∗.
+(A3)
+In the remaining of this contribution we will almost systematically take (A1)-(A2)-(A3) as
+assumptions. We do not require that AΩ×Γ = A∗
+Ω×Γ. Let us underline that the assumptions
+above are fulfilled by AΩ, AΓ, AΩ×Γ if and only if they are fulfilled by A∗
+Ω, A∗
+Γ, A∗
+Ω×Γ (recall
+that adjunction does not involve any complex conjugation here). The last hypothesis in (A3)
+implies (see e.g. [2, Thm.2.19])
+range(AΩ×Γ) = ker(A∗
+Ω×Γ)◦.
+(16)
+hence codim(range(AΩ×Γ)) = dim(ker(A∗
+Ω×Γ)). The source functional in (14) is assumed to
+take the similar form ⟨ℓΩ×Γ, (v, q)⟩ := ⟨ℓΩ, v⟩+⟨ℓΓ, (v|Γ, q)⟩ where ⟨ℓΩ, v⟩ :=
+�
+Ω fv dx for some
+f ∈ L2(Ω) and ℓΓ ∈ Hb(Γ)∗ = H−1/2(Γ)×H+1/2(Γ) is chosen in accordance with the boundary
+condition.
+Now we consider concrete boundary conditions, exhibit corresponding appropriate choices of
+AΓ and point how these situations fit the previous assumptions (A1)-(A2)-(A3). Here and in
+the following, for the sake of conciseness, we shall take the notational convention (see (11)),
+TΓ := TRd\Ω.
+Example 3.1 (Dirichlet boundary condition). In the case of a Dirichlet boundary condi-
+tion, we set AΓ(α, p) := (p, α) and ℓΓ := (0, g) for some g ∈ H1/2(Γ). We have ℑm{⟨AΓ(u), u⟩} =
+0 for all u, which fits i) of (A3). Formulation (14) reduces to a variational formulation of a
+Helmholtz problem with a Dirichlet condition imposed by means of a Lagrange parameter at
+the boundary
+u ∈ H1(Ω), p ∈ H−1/2(Γ) such that
+�
+Ω µ−1∇u · ∇v − κ2uv dx +
+�
+Γ pv dσ =
+�
+Ω fvdx
+∀v ∈ H1(Ω),
+�
+Γ uq dσ =
+�
+Γ gq dσ
+∀q ∈ H−1/2(Γ).
+6
+
+Whenever there is existence and uniqueness of the solution pair (u, p) then p = −nΩ · ∇u|Γ.
+Conditions in (A2) guarantee that the volume part of this equation is coercive modulo the
+compact term attached to κ. Hence the operator associated to this system is of Fredholm type
+with index 0. In particular it has closed range, which fits ii) of (A3).
+Example 3.2 (Neumann boundary condition). In the case of Neumann conditions, the
+boundary data is g ∈ H−1/2(Γ) and we choose AΓ(α, p) := (0, T−1
+Γ p) and ℓΓ := (g, 0). Again
+we have ℑm{⟨AΓ(u), u⟩} = 0 for all u, so this choice also matches i) of (A3). The primary
+formulation (14) writes
+u ∈ H1(Ω), p ∈ H−1/2(Γ) such that
+�
+Ω µ−1∇u · ∇v − κ2uv dx =
+�
+Ω fvdx +
+�
+Γ gvdσ
+∀v ∈ H1(Ω),
+�
+Γ qT−1
+Γ p dσ = 0
+∀q ∈ H−1/2(Γ),
+(17)
+where u is decoupled from p. Actually we have in particular p = 0 and this variable is not
+supposed to receive any particular interpretation.
+Since T−1
+Γ
+: H−1/2(Γ) → H1/2(Γ) is an
+isomorphism, the operator AΩ×Γ associated to (17) is of Fredholm type with index 0.
+Example 3.3 (Robin boundary condition). Consider a bounded linear map Λ : H1/2(Γ) →
+H−1/2(Γ) that satisfies ℜe{⟨Λ(v), v⟩} > 0 ∀v ∈ H1/2(Γ)\{0} (as a typical example: Λ(v) = λv
+with λ > 0). In this case again the boundary data is g ∈ H−1/2(Γ) and we choose AΓ(α, p) :=
+(−iΛα, T−1
+Γ p) and ℓΓ := (g, 0).
+This choice of AΓ corresponds to the boundary condition
+nΩ · ∇u|Γ − iΛ(u) = 0 on Γ. Formulation (14) writes
+u ∈ H1(Ω), p ∈ H−1/2(Γ) such that
+�
+Ω µ−1∇u · ∇v − κ2uv dx − i
+�
+Γ vΛ(u)dσ =
+�
+Ω fvdx +
+�
+Γ gvdσ
+∀v ∈ H1(Ω)
+�
+Γ qT−1
+Γ p dσ = 0
+∀q ∈ H−1/2(Γ)
+which is a variant of (17) involving i
+�
+Γ vΛ(u)dσ as an additional term. Again p is decoupled
+from the rest of the system and p = 0. Again the operator AΩ×Γ associated to this system is
+of Fredholm type with index 0.
+4
+Multi-domain functional setting
+The boundary value problem (1) has been reformulated as an equivalent global variational
+problem with (14). As we aim at extending an analytical framework for domain decomposition
+by substructuration though, we are going to reshape Formulation (14), adapting it to a multi-
+domain geometrical configuration. For this, we need to introduce notations adapted to domain
+decomposition. Consider a decomposition into a collection of non-overlapping Lipschitz open
+sets Ωj ⊂ Rd, j = 1, . . . , J that satisfy
+Ω = Ω1 ∪ · · · ∪ ΩJ,
+with Ωj ∩ Ωk = ∅ for j ̸= k.
+(18)
+Such a decomposition may very well admit a non-trivial wire-basket i.e.
+the set of cross
+points is non-empty, and we wish to underline that this situation is covered by the subsequent
+analysis. We shall refer to the skeleton of the decomposition by
+Σ := ∂Ω1 ∪ · · · ∪ ∂ΩJ.
+(19)
+7
+
+Note that Γ = ∂Ω ⊂ Σ. We need to introduce notations for function spaces adapted to this
+multi-domain setting. In this context, cartesian product spaces are probably the most natural,
+so we set
+Hb(Γ) := H
+1
+2 (Γ) × H− 1
+2(Γ)
+H(Ω) := Hb(Γ) × H1(Ω1) × · · · × H1(ΩJ)
+H(Σ) := H
+1
+2 (Γ) × H
+1
+2(∂Ω1) × · · · × H
+1
+2(∂ΩJ)
+(20)
+As cartesian products, these spaces are equipped with norms and duality pairings given by
+(7). Apart from the boundary terms attached to Hb(Γ), the space H(Ω) should be understood
+as functions defined over Ω, admitting potential jumps through interfaces. The space H(Σ)
+consists in tuples of Dirichlet traces. Its dual is
+H(Σ)∗ = H− 1
+2 (Γ) × H− 1
+2(∂Ω1) × · · · × H− 1
+2(∂ΩJ).
+We need to introduce several operators acting in these spaces. First we shall consider the
+operator T : H(Σ) → H(Σ)∗ defined as the block diagonal operator acting locally in each
+subdomain
+T := diag(TΓ, TΩ1, . . . , TΩJ)
+where TΓ := TR\Ω
+(21)
+and each TΩj is defined with (11). The norms ∥ · ∥T and ∥ · ∥T−1 defined by (6) and (21) are
+equivalent to ∥ · ∥H(Σ) and ∥ · ∥H(Σ)∗, which stems from the analogous property being satisfied
+locally by each TΩj. These norms will play an important role in the subsequent analysis. Next
+we introduce a boundary trace operator B : H(Ω) → H(Σ) and defined by
+B := diag(BΓ, BΩ1, . . . , BΩJ)
+where BΓ(α, p) := α
+(22)
+and each BΩj is the Dirichlet trace operator interior to subdomain Ωj as defined in (9). By
+definition of T we have ∥B(u)∥T ≤ ∥u∥H(Ω) for all u ∈ H(Ω), since a similar inequality
+is satisfied in each subdomain locally according to (13). We can also form a multi-domain
+harmonic lifting map B† : H(Σ) → H(Ω) defined as the block-diagonal operator as follows
+B† = diag(B†
+Γ, B†
+Ω1, . . . , B†
+ΩJ)
+where B†
+Γ(α) := (α, 0)
+(23)
+and each B†
+Ωj as defined in (10).
+With this definition we have BB† = Id and B†B is an
+orthogonal projector in H(Ω). Finally we also need to consider a restriction operator R :
+H(Ω×Γ) → H(Ω) that embeds pairs (u, p) ∈ H(Ω×Γ) = H1(Ω)×H−1/2(Γ) into the cartesian
+product H(Ω) by restricting locally to each subdomain
+R(u, p) := ((u|Γ, p), u|Ω1, . . . , u|ΩJ)
+for u ∈ H1(Ω), p ∈ H−1/2(Γ).
+(24)
+The image of this operator range(R) = R(H(Ω×Γ)) is a particular subspace of H(Ω) spanned
+by tuples of functions that match through interfaces. This matching property is precisely
+8
+
+what characterizes Dirichlet transmission conditions through interfaces of the decomposition
+(18). This is why we dedicate notations to this.
+X(Ω) := {R(u, p), u ∈ H1(Ω), p ∈ H−1/2(Γ)}
+X(Σ) := {B(u), u ∈ X(Ω)}
+X(Σ)◦ := {p ∈ H(Σ)∗, ⟨p, v⟩ = 0 ∀v ∈ X(Σ)}.
+(25)
+A rapid inspection of the previous definitions shows that X(Σ) = {(u|Γ, u|∂Ω1, . . . , u|∂ΩJ), u ∈
+H1(Ω)} i.e. these are the tuples of Dirichlet traces that match through interfaces. The space
+X(Σ) (resp.
+X(Ω)) is a closed subspace of H(Σ) (resp.
+H(Ω)) that encodes the Dirichlet
+transmission conditions through interfaces, while X(Σ)◦ is a closed subspace of H(Ω)∗ that
+encodes the Neumann transmission conditions. Indeed, considering restriction to interfaces in
+the sense of distributions,
+(v0, . . . , vJ) ∈ X(Σ)◦ =⇒ vj = +vk on Γj ∩ Γk,
+(p0, . . . , pJ) ∈ X(Σ)◦ =⇒ pj = −pk on Γj ∩ Γk.
+(26)
+It is clear from these definitions that X(Ω) = {u ∈ H(Ω), B(u) ∈ X(Σ)}.
+In particular
+ker(B) ⊂ X(Ω). Recall the definition of polar sets given by (3). The following lemma is a
+continuous counterpart to [6, Lem.2.1].
+Lemma 4.1.
+i) ker(B)◦ = range(B∗)
+ii) ker(B∗) = {0}
+iii) X(Ω) = B−1(X(Σ))
+iv) X(Ω)◦ = B∗(X(Σ)◦)
+Proof:
+The first and second results are direct consequences of the surjectivity of the trace map
+B : H(Ω) → H(Σ) combined with Theorem 4.7, 4.12 and 4.15 of [22]. The third result is a
+rephrasing of X(Ω) = {u ∈ H(Ω), B(u) ∈ X(Σ)} in condensed form. To prove the last result,
+first observe that B∗(X(Σ)◦) ⊂ X(Ω)◦ by routine verifications.
+Now pick an arbitrary p ∈ X(Ω)◦. Since ker(B) ⊂ X(Ω) ⇒ X(Ω)◦ ⊂ ker(B)◦ = range(B∗),
+there exists q ∈ H(Σ)∗ such that p = B∗q. For any v ∈ X(Σ), there exists u ∈ X(Ω) such
+that v = B(u), which implies that ⟨q, v⟩ = ⟨p, u⟩ = 0. From this we conclude that q ∈ X(Σ)◦
+hence p ∈ B∗(X(Σ)◦), which proves X(Ω)◦ ⊂ B∗(X(Σ)◦).
+□
+In Item iii) of the lemma above, B−1(X(Σ)) = {u ∈ H(Ω), B(u) ∈ X(Σ)} refers to a pre-image
+(the operator B is obviously non-invertible i.e.
+ker(B) ̸= {0}).
+The following orthogonal
+decomposition was established in [17, Prop.4.2].
+Proposition 4.2.
+We have H(Σ)∗ = X(Σ)◦ ⊕ T(X(Σ)) and this decomposition is T−1-orthogonal.
+The orthogonal decomposition of the previous result can be used to elaborate a characteriza-
+tion of transmission conditions. The following result was established in [17, Prop.5.4].
+9
+
+Proposition 4.3.
+Let Q : H(Σ)∗ → H(Σ)∗ refer to the T−1-orthogonal projection onto T(X(Σ)).
+Then the
+operator Π := 2Q − Id is a T−1-isometric involution i.e. Π2 = Id, ∥Π(q)∥T−1 = ∥q∥T−1 for
+all q ∈ H(Σ)∗. Moreover, for any pair (u, p) ∈ H(Σ) × H(Σ)∗, we have
+(u, p) ∈ X(Σ) × X(Σ)◦
+⇐⇒
+−p + iT(u) = Π(p + iT(u)).
+(27)
+The characterization above relies on an exchange operator Π which is characteristic of Opti-
+mized Schwarz Methods (OSM, see e.g. [1, Eq.37]) and ultra-weak variational formulations
+(UWVF) see e.g. [3, Eq.1.19]. An explicit expression of this operator in terms of double layer
+potentials attached to the equation −∆ + γ−2 was provided in [5, §5.2].
+5
+Multi-domain variational formulation
+Using the notations introduced in the previous sections, we now rewrite the primary formula-
+tion (14), decomposing it according to the subdomain partition (18). Pick u, v arbitrarily in
+H1(Ω) and expand the integral coming into play in the definition (A2) of AΩ. This leads to
+⟨AΩu, v⟩ = ⟨AΩ1(u|Ω1), v|Ω1⟩ + · · · + ⟨AΩJ(u|ΩJ), v|ΩJ⟩
+with
+⟨AΩju, v⟩ :=
+�
+Ωj
+µ−1∇u · ∇v − κ2uv dx
+(28)
+In the expression above only u|Ωj, v|Ωj ∈ H1(Ωj) come into play in the term attached to
+Ωj. The source term in (14) can be decomposed in a similar manner ℓΩ(v) = ℓΩ1(v|Ω1) +
+. . . ℓΩJ(v|ΩJ). The above decompositions lead to introducing a block-diagonal operator A :
+H(Ω) → H(Ω)∗ associated to these local bilinear forms i.e. defined by
+A := diag(AΓ, AΩ1, . . . , AΩJ)
+so that AΩ×Γ = R∗AR.
+(29)
+We have factorized the operator of our primary boundary value problem AΩ×Γ, and this
+factorization is interesting from the perspective of domain decomposition because local sub-
+problems are disconnected from one another in A. The following property is inherited from
+the assumptions we made in §3 about AΩ×Γ, µ, κ and AΓ,
+ℑm{⟨A(u), u⟩} ≤ 0
+∀u ∈ H(Σ).
+(30)
+We also need a unique solvability property for local problems with impedance boundary con-
+dition. Because we do not make much specific assumptions regarding the boundary operator
+AΓ, we take this further property as an assumption:
+Assumption:
+A − iB∗TB : H(Ω) → H(Ω)∗
+is an isomorphism.
+(A4)
+A notable consequence of (A2), (A3) and (A4) is that ker(A) ∩ ker(B) = {0}. Since A, T and
+B are subdomain-wise block-diagonal, the assumption above is actually equivalent to imposing
+that each AΩj − iB∗
+ΩjTΩjBΩj : H(Ωj) → H(Ωj)∗ and AΓ − iB∗
+ΓTΓBΓ : Hb(Γ) → Hb(Γ)∗ are
+10
+
+isomorphisms.
+These conditions are fulfilled in many concrete circumstances.
+As regards
+interior contributions, for example, we have the following simple consequence of the unique
+continuation principle.
+Lemma 5.1.
+Assume (A1)-(A2) and that µ, κ are constants (i.e.
+do not depend on x).
+Then for any
+j = 1, . . . , J the operator AΩj − iB∗
+ΩjTΩjBΩj : H(Ωj) → H(Ωj)∗ is an isomorphism.
+Proof:
+Let us denote ω = Ωj for the sake of conciseness. According to (A2), there exists α > 0
+such that
+α∥u∥2
+H1(ω) ≤ ℜe{⟨˜Aω(u), u⟩}
+∀u ∈ H1(ω),
+⟨˜Aω(u), v⟩ := ⟨(Aω − iB∗
+ωTωBω)u, v⟩ +
+�
+ω(1 + κ2)uvdx.
+Applying Lax-Milgram’s lemma, we see that the operator ˜Aω : H(ω) → H(ω)∗ is an isomor-
+phism hence, since it differs by a compact perturbation, that Aω−iB∗
+ωTωBω is of Fredholm type
+with index 0, see e.g. [17, Chap.2]. There only remains to prove that ker(Aω − iB∗
+ωTωBω) =
+{0}. Pick any u ∈ H1(ω) such that (Aω − iB∗
+ωTωBω)u = 0. Then we have
+∥Bω(u)∥2
+Tω ≤ −ℑm{⟨(Aω − iB∗
+ωTωBω)u, u⟩} = 0.
+From this we conclude that u|∂ω = Bω(u) = 0 hence Aω(u) = 0. On the other hand Aω(u) =
+0 ⇒ nω · ∇u|∂ω = 0. There only remains to apply the unique continuation principle, see e.g.
+Lemma 2.2 in [24], to conclude that u = 0 in ω.
+□
+Regarding classical boundary conditions and the associated choice of AΓ, we can also examine
+the invertibility of AΓ − iB∗
+ΓTΓBΓ.
+Example 5.2 (Dirichlet condition). Taking the same notations as in Example 3.1, in this
+situation we have the following expression (AΓ−iB∗
+ΓTΓBΓ)(α, p) = (p−iTΓα, α). We conclude
+that AΓ − iB∗
+ΓTΓBΓ is continuously invertible with
+(AΓ − iB∗
+ΓTΓBΓ)−1(p, α) = (α, p + iTΓα).
+Example 5.3 (Neumann condition). Taking the same notations as in Example 3.2, we have
+(AΓ − iB∗
+ΓTΓBΓ)(α, p) = (−iTΓα, T−1
+Γ p). We conclude that AΓ − iB∗
+ΓTΓBΓ is continuously
+invertible with
+(AΓ − iB∗
+ΓTΓBΓ)−1(p, α) = (iT−1
+Γ p, TΓα).
+Example 5.4 (Robin condition). Taking the same notations as in Example 3.3, we have
+(AΓ−iB∗
+ΓTΓBΓ)(α, p) = (−i(Λ+TΓ)α, T−1
+Γ p). Because ℜe{⟨Λ(α), α⟩} > 0 for all α ∈ H1/2(Γ),
+we see that Λ+TΓ is coercive hence invertible and AΓ−iB∗
+ΓTΓBΓ is then continuously invertible
+with
+(AΓ − iB∗
+��TΓBΓ)−1(p, α) = (i(Λ + TΓ)−1p, TΓα).
+Similarly to what precedes, define ℓ ∈ H(Ω)∗ by ⟨ℓ, v⟩ = ℓΓ(v0, q) + ℓΩ1(v1) + · · · + ℓΩJ(vJ),
+and we have ℓΩ×Γ = R∗ℓ. The primary variational problem (14) can then rewritten by means
+of A as follows: find u ∈ H(Ω × Γ) such that ⟨AR(u), R(v)⟩ = ⟨ℓ, R(v)⟩ for all v ∈ H(Ω × Γ).
+Making use of the definition of X(Ω) as the image of R see (25), this also rewrites
+u ∈ X(Ω) and
+⟨A(u), v⟩ = ⟨ℓ, v⟩ ∀v ∈ X(Ω).
+(31)
+11
+
+6
+Closed linear manifolds interpretation
+Formulation (14) which is the starting point of this study, is not assumed to be a priori uniquely
+solvable. The kernel of AΩ×Γ might be non-trivial. In many relevant applications though, it is
+of Fredholm type, and this is why we are interested in studying how this Fredholmness carries
+over in the multi-domain context.
+For this we are going to consider the skew-symmetric
+bilinear form [·, ·] : ( H(Σ) × H(Σ)∗)2 → C defined by
+[(u, p), (v, q)] := ⟨u, q⟩ − ⟨v, p⟩
+u, v ∈ H(Σ), p, q ∈ H(Σ)∗.
+(32)
+This form is obviously non-degenerate and can be used as a duality pairing over the space of
+tuples of Dirichlet-Neumann pairs of traces. Indeed denote
+H (Σ) := H(Σ) × H(Σ)∗
+with norm:
+∥(v, q)∥2
+T×T−1 := ∥v∥2
+T + ∥q∥2
+T−1
+then for any ϕ ∈ H (Σ)∗, there exists a unique u ∈ H (Σ) such that [u, v] = ϕ(v) ∀v ∈ H (Σ).
+In other words, the pairing (32) puts H (Σ) in self-duality. We now introduce the subspace
+of so-called Cauchy data that directly relates to the boundary value problem under study,
+C (A) := {(B(u), p) | (u, p) ∈ H(Ω) × H(Σ)∗, Au = B∗p}
+(33)
+It must be understood as the space of tuples of Dirichlet-Neumann trace pairs stemming from
+solutions to the problems local to each subdomain. If A : H(Ω) → H(Ω)∗ is an isomorphism,
+we can define the associated Neumann-to-Dirichlet operator NtDA := BA−1B∗ and then
+C (A) := {(NtDA(p), p) | p ∈ H(Σ)∗} appears to be the graph of it. On the other hand C (A)
+is properly defined even if A fails to be invertible.
+Lemma 6.1.
+Assume (A1)-(A2)-(A3)-(A4). The application (v, p) → p − iT(v) continuously and isomor-
+phically maps C (A) into H(Σ)∗ and, for all (v, p) ∈ C (A), satisfies the estimates
+∥v∥2
+T + ∥p∥2
+T−1 ≤ ∥p − iTv∥2
+T−1
+1
+2∥p − iTv∥2
+T−1 ≤ ∥v∥2
+T + ∥p∥2
+T−1.
+Proof:
+It suffices to prove surjectivity and the estimates. To prove surjectivity, pick an arbitrary
+q ∈ H(Σ)∗ and define u = (A − iB∗TB)−1B∗q. The pair (v, p) = (B(u), q + iTB(u)) satisfies
+Au = B∗p so that (v, p) ∈ C (A) and, by construction, we have p − iTv = q.
+To prove the estimates, pick an arbitrary pair (v, p) ∈ C (A). According to (33) there exists
+u ∈ H(Ω) such that B(u) = v and A(u) = B∗(p), hence ⟨p, v⟩ = ⟨p, B(u)⟩ = ⟨B∗(p), u⟩ =
+⟨A(u), u⟩. Taking account of (30), we deduce 0 ≤ ℜe{i⟨p, v⟩} ≤ ∥v∥2
+T +∥p∥2
+T−1 and conclude
+0 ≤ ∥p − iTv∥2
+T−1 − (∥v∥2
+T + ∥p∥2
+T−1) ≤ ∥v∥2
+T + ∥p∥2
+T−1.
+□
+In the previous lemma, the space of Cauchy data has been proven boundedly isomorphic to a
+Hilbert space and, as such, is closed.
+12
+
+Corollary 6.2.
+Assume (A1)-(A2)-(A3)-(A4). The subspace C (A) is closed in H (Σ).
+The space of Cauchy data can be complemented in various ways. The next proposition exhibits
+one possibility.
+Proposition 6.3.
+Assume (A1)-(A2)-(A3)-(A4). Define G (iT) := {(v, iT(v)), v ∈ H(Σ)}. Then
+H (Σ) = C (A) ⊕ G (iT).
+Proof:
+First of all, assume that (u, p) ∈ C (A) ∩ G (iT). This that there exists v ∈ H(Ω) such
+that Av = B∗p and Bv = u, and that p = iTu. Combining these equations yields (A −
+iB∗TB)v = 0 hence v = 0 according to Lemma 5.1, and finally (u, p) = 0. We have proved
+that C (A) ∩ G (iT) = {0}.
+Now take an arbitrary (u, p) ∈ H(Σ) × H(Σ)∗. Since B : H(Ω) → H(Σ) is surjective, there
+exists w ∈ H(Ω) such that B(w) = u. Define v ∈ H(Ω) by v = (A − iB∗TB)−1(Aw − B∗p)
+which is valid a definition since A − iB∗TB : H(Ω) → H(Ω)∗ is an isomorphism according to
+Lemma 5.1. We have in particular A(w − v) = B∗(p − iTBv). Set
+u1 = B(v),
+p1 = iTu1 = iTB(v),
+u2 = B(w − v) = u − u1,
+p2 = p − iTBv = p − p1.
+(34)
+By construction we have (u1, p1) ∈ G (iT). Moreover B(w −v) = u2 and A(w −v) = B∗p2 so
+that (u2, p2) ∈ C (A). Finally, the second line in (34) indicates that (u, p) = (u1, p1)+(u2, p2)
+which thus proves (u, p) ∈ C (A) + G (iT). We have just established that C (A) + G (iT) =
+H(Σ) ⊕ H(Σ)∗ which ends the proof.
+□
+The space G (iT) is simply the graph of the (bounded) operator iT : H(Σ) → H(Σ)∗. In the
+present analysis, it plays a secondary role and shall be used only to prove results about C (A).
+We have the following immediate result.
+Lemma 6.4.
+Define G (iT)♯ := {u ∈ H (Σ), [u, v] = 0 ∀v ∈ G (iT)}. Then G (iT)♯ = G (iT).
+The proof is definitely straightforward. This result means that G (iT) is its own polar set
+under the pairing [·, ·]. As we see now, the space C (A) fulfills a similar property.
+Proposition 6.5.
+Assume (A1)-(A2)-(A3)-(A4). Define C (A)♯ := {u ∈ H (Σ), [u, v] = 0 ∀v ∈ C (A)}. Then
+C (A)♯ = C (A∗).
+Proof:
+First of all we have C (A∗) ⊂ C (A)♯. Indeed take any (u, p) ∈ C (A). By definition, there
+exists w ∈ H(Ω) such that B(w) = u and Aw = B∗p. Then for any (u′, p′) ∈ C (A∗), since
+B(w′) = u′ and A∗w′ = B∗p′ for some w′ ∈ H(Ω), we have
+[(u, p), (u′, p′)] = ⟨u, p′⟩ − ⟨u′, p⟩ = ⟨B(w), p′⟩ − ⟨B(w′), p⟩
+= ⟨w, B∗(p′)⟩ − ⟨w′, B∗(p)⟩
+= ⟨w, A∗(w′)⟩ − ⟨w′, A(w)⟩ = 0.
+13
+
+Hence, to finish the proof, we need to show that C (A)♯ ⊂ C (A∗). For that, pick an arbitrary
+u = (u, p) ∈ C (A)♯. The hypothesis of Section 3 hold for A∗
+Ω×Γ instead of AΩ×Γ, hence we
+can apply Proposition 6.3 to A∗. This yields a decomposition u = u1 +u2 for some u1 ∈ C (A∗)
+and some u2 ∈ G (iT). We have to prove that u2 = 0. By assumption we have
+0 = [u, v] = [u1, v] + [u2, v] = [u2, v]
+∀v ∈ C (A),
+since C (A) ⊂ C (A∗)♯. Next Lemma 6.4 implies that 0 = [u2, v] = [u2, v + v′] for all v ∈ C (A)
+and all v′ ∈ G (iT). Since C (A) ⊕ G (iT) = H (Σ) according to Proposition 6.3, we conclude
+that 0 = [u2, w] ∀w ∈ H (Σ) hence finally u2 = 0. This shows that u = u1 ∈ C (A∗). We have
+just established that C (A)♯ ⊂ C (A∗).
+□
+We point that, because C (A) is closed, the previous result also implies that C (A) = C (A∗)♯.
+Self-polarity appears to be a property of the following subspace (see Proposition 4.3) that is
+pivotal in characterizing transmission conditions
+X (Σ) := X(Σ) × X(Σ)◦.
+Indeed we have X (Σ) = X (Σ)♯ := {u ∈ H (Σ), [u, v] = 0 ∀v ∈ X (Σ)} by the very definition
+of X (Σ), as X(Σ)◦◦ = X(Σ) since X(Σ) is a closed subspace of H(Σ) (see e.g. [22, Thm.4.7]
+or [2, Prop.1.9]). The next result establishes an important connection between the two spaces
+C (A), X (Σ) and our primary boundary value problem (14).
+Proposition 6.6.
+Assume (A1)-(A2)-(A3)-(A4).
+The operator u �→ (BR(u), (B†)∗AR(u)) continuously and
+isomorphically maps ker(AΩ×Γ) onto C (A) ∩ X (Σ). As a consequence
+dim(ker(AΩ×Γ) ) = dim(C (A) ∩ X (Σ)).
+Proof:
+Let u ∈ H(Ω×Γ) satisfy AΩ×Γ(u) = 0. In particular R(u) ∈ X(Ω) and AR(u) ∈ X(Ω)◦, see
+(24) and (31). According to iv) of Lemma 4.1, there exists p ∈ X(Σ)◦ such that AR(u) = B∗p
+and it is unique since B∗ : H(Σ)∗ → H(Ω)∗ is injective. We have
+(B†)∗AR(u) = (B†)∗B∗p = (BB†)∗p = p.
+Setting v := B · R(u), by construction (v, p) ∈ C (A).
+We also have v ∈ X(Σ) since
+R(u) ∈ X(Ω), so that (v, p) ∈ X(Σ) × X(Σ)◦ = X (Σ). In addition, the formula (v, p) =
+(BRu, (B†)∗ARu) establishes the continuous dependency of (v, p) on u.
+Reciprocally, consider an arbitrary pair (v, p) ∈ C (A) ∩ X (Σ). Since (v, p) ∈ C (A),
+there exists w ∈ H(Ω) such that Aw = B∗p and B(w) = v, and such a w is unique since
+ker(A)∩ker(B) = {0}, according to Lemma 5.1. As v ∈ X(Σ), we have w ∈ X(Ω) = B−1(X(Σ))
+according to iii) of Lemma 4.1, so there exists u ∈ H(Ω × Γ) such that R(u) = w and such
+a u is unique due to the injectivity of R : H(Ω × Γ) → H(Ω). This leads to AR(u) = B∗p
+and p ∈ X(Σ)◦ ⇒ B∗p ∈ X(Ω)◦ = ker(R∗). Since X(Ω) = R(H(Ω × Γ)), we conclude that
+0 = R∗AR(u) = AΩ×Γ(u).
+□
+Lemma 6.7.
+Assume (A1)-(A2)-(A3)-(A4). The operator (u, p) �→ R∗(B∗p − AB†u) continuously maps
+(C (A∗) ∩ X (Σ))♯ into range(AΩ×Γ).
+14
+
+Proof:
+Take an arbitrary (u, p) ∈ (C (A∗) ∩ X (Σ))♯ and set f = R∗(B∗p − AB†u).
+Ap-
+plying Proposition 6.6 to A∗
+Ω×Γ instead of AΩ×Γ shows that ϕ ∈ ker(A∗
+Ω×Γ) ⇒ (v, q) =
+(BR(ϕ), (B†)∗A∗R(ϕ)) ∈ C (A∗) ∩ X (Σ). Hence ⟨f, ϕ⟩ = ⟨R∗(B∗p − AB†u), ϕ⟩ = ⟨p, BRϕ⟩ −
+⟨u, (B†)∗A∗Rϕ⟩ = [(v, q), (u, p)] = 0. This proves f ∈ ker(A∗
+Ω×Γ)◦ = range(AΩ×Γ) according
+to (16).
+□
+Proposition 6.8.
+Assume (A1)-(A2)-(A3)-(A4). Then C (A) + X (Σ) = (C (A∗) ∩ X (Σ))♯. In particular the
+subspace C (A) + X (Σ) is closed in H (Σ).
+Proof:
+Clearly we have C (A) + X (Σ) ⊂ (C (A∗) ∩ X (Σ))♯, so we only need to establish that
+(C (A∗) ∩ X (Σ))♯ ⊂ C (A) + X (Σ). Pick any pair (pd, pn) ∈ (C (A∗) ∩ X (Σ))♯. According to
+Lemma 6.7 we have R∗(B∗pn − AB†pd) ∈ range(AΩ×Γ). Applying the definition of A given
+by (29), there exists ϕ ∈ X(Ω) satisfying ⟨Aϕ, w⟩ = ⟨B∗pn − AB†pd, w⟩ for all ∀w ∈ X(Ω).
+Set φ = ϕ + B†(pd) and ud = B(φ) = B(ϕ) + pd.
+By construction, ⟨A(φ), w⟩ =
+⟨pn, B(w)⟩ = 0 ∀w ∈ ker(B) ⊂ X(Ω), which rewrites A(φ) ∈ ker(B)◦. Applying i) of Lemma
+4.1 we have Aφ = B∗un for some un ∈ H(Σ)∗. This implies in particular un = (BB†)∗un =
+(B†)∗B∗un = (B†)∗Aφ.
+We have Aφ = B∗un and Bφ = ud hence (ud, un) ∈ C (A). On the other hand pd −
+ud = −Bϕ ∈ X(Σ) since ϕ ∈ X(Ω) and, for any w ∈ X(Σ) we have B†(w) ∈ X(Ω) hence
+⟨pn−un, w⟩ = ⟨Aφ, B†w⟩−⟨Aφ, B†w⟩ = 0, which implies pn−un ∈ X(Σ)◦. Finally (ud, un) ∈
+C (A) and (pd, pn) − (ud, un) ∈ X (Σ) imply that (pd, pn) ∈ C (A) + X (Σ).
+□
+Corollary 6.9.
+Assume (A1)-(A2)-(A3)-(A4). Then
+codim(C (A) + X (Σ) ) = codim(range(AΩ×Γ) ).
+Proof:
+We have (C (A) + X (Σ))♯ = C (A)♯ ∩ X (Σ)♯ see e.g.
+[2, Prop.2.14].
+According to
+Proposition 6.5 applied to A∗, and since X (Σ)♯ = X (Σ) by construction, we conclude
+that (C (A) + X (Σ))♯ = C (A∗) ∩ X (Σ). As the bilinear pairing [·, ·] is non-degenerate and
+C (A) + X (Σ) is closed according to Proposition 6.8, we conclude codim(C (A) + X (Σ)) =
+dim((C (A) + X (Σ))♯) = dim(C (A∗) ∩ X (Σ)). There only remains to apply Proposition 6.6
+to A∗
+Ω×Γ combined with (16).
+□
+7
+Scattering operator
+Proposition 6.6 and 6.8 and Corollary 6.9 above show that the kernel and the range of AΩ×Γ
+are closely related to the pair of subspaces C (A), X (Σ). This can be exploited to study other
+formulations of the same boundary value problem.
+Proposition 7.1.
+Assume (A1)-(A2)-(A3)-(A4). If u ∈ X(Ω) satisfies (31), then there exists a unique p ∈ H(Σ)∗
+such that the pair (u, p) satisfies
+u ∈ H(Ω), p ∈ H(Σ)∗,
+Au − B∗p = ℓ,
+− p + iTBu = Π(p + iTBu).
+(35)
+15
+
+Reciprocally if the pair (u, p) ∈ H(Ω) × H(Σ)∗ satisfies (35), then u satisfies (31).
+Proof:
+Assume first that u ∈ X(Ω) satisfies (31). This formulation rewrites equivalently as Au −
+ℓ ∈ X(Ω)◦. Since X(Ω)◦ = B∗(X(Σ)◦) according to iv) Lemma 4.1, and as B∗ : H(Σ)∗ → H(Ω)∗
+is injective (B is surjective), there exists a unique p ∈ X(Σ)◦ such that Au − ℓ = B∗p. On
+the other hand, u ∈ X(Ω) ⇒ B(u) ∈ X(Σ) according to iii) of Lemma 4.1. Finally applying
+Proposition 4.3, we obtain −p + iTBu = Π(p + iTBu).
+Reciprocally, assume that (35) holds. Then, according to Proposition 4.3, we have p ∈
+X(Σ)◦ and B(u) ∈ X(Σ). Moreover we have B(u) ∈ X(Σ) ⇒ u ∈ X(Ω) according to iii)
+of Lemma 4.1. Since p ∈ X(Σ)◦, we have B∗p ∈ X(Ω)◦ so that, for any v ∈ X(Ω) we have
+0 = ⟨B∗p, v⟩ = ⟨Au − ℓ, v⟩. To sum up, we have proved that u ∈ X(Ω) and ⟨Au, v⟩ =
+⟨ℓ, v⟩ ∀v ∈ X(Ω).
+□
+In a domain decomposition context, a substructuring strategy applied to Problem (14) nat-
+urally leads to eliminating the volume unknowns in (35). This is performed by means of a
+scattering map that takes ingoing traces as input and returns outgoing traces as output.
+Proposition 7.2.
+Assume (A1)-(A2)-(A3)-(A4). There exists a unique bounded linear map S : H(Σ)∗ → H(Σ)∗,
+later referred to as scattering operator, satisfying
+p + iTv = S(p − iTv)
+∀(v, p) ∈ C (A).
+(36)
+It is also given by the formula S = Id+ 2iTB(A − iB∗TB)−1B∗. It is T−1-contractive and, for
+any q ∈ H(Σ)∗, satisfies
+∥S(q)∥2
+T−1 + 4|ℑm{⟨A(u), u⟩}| = ∥q∥2
+T−1
+where u = (A − iB∗TB)−1B∗q.
+Proof:
+We follow the proof pattern presented e.g. in [6, Lem.5.2]. First of all, Identity (36) clearly
+and unambiguously defines the operator S as a linear map according to Lemma 6.1. Next,
+pick an arbitrary q ∈ H(Σ)∗ and set u = (A − iB∗TB)−1B∗q and p = q + iTB(u). We have
+Au − B∗p = 0 and q = p − iTB(u) and S(q) = p + iTB(u) = q + 2iTB(u), which leads
+to S(q) = (Id + 2iTB(A − iB∗TB)−1B∗)q. Finally developing the squared norm, and taking
+account of (30), we have
+∥S(q)∥2
+T−1 = ∥p + iTB(u)∥2
+T−1
+= ∥p − iTB(u)∥2
+T−1 + 4ℑm{⟨q, B(u)⟩} + 4∥B(u)∥2
+T
+= ∥q∥2
+T−1 + 4ℑm{⟨B∗(q), u⟩} + 4∥B(u)∥2
+T
+= ∥q∥2
+T−1 + 4ℑm{⟨A(u), u⟩} − 4ℑm{i⟨B∗TB(u), u⟩} + 4∥B(u)∥2
+T
+= ∥q∥2
+T−1 − 4|ℑm{⟨A(u), u⟩}|
+□
+The space of Cauchy data was used to characterize the scattering operator. Reciprocally, the
+scattering operator provides a characterization of the space of Cauchy data. The following
+result should be compared with (27).
+16
+
+Lemma 7.3.
+Assume (A1)-(A2)-(A3)-(A4). For any (v, p) ∈ H (Σ) we have:
+(v, p) ∈ C (A) ⇐⇒ p + iTv = S(p − iTv).
+Proof:
+From the very definition of the scattering operator in Proposition 7.2, it is clear that
+(v, p) ∈ C (A) ⇒ p + iTv = S(p − iTv). Reciprocally pick arbitrarily some (v, p) ∈ H (Σ)
+such that p + iTv = S(p − iTv). We know from Proposition 6.3 that there exists v′ ∈ H(Σ)
+such that (v − v′, p − iTv′) ∈ C (A) so applying Proposition 7.2 we obtain
+(p − iTv′) + iT(v − v′) = S( (p − iTv′) − iT(v − v′) )
+⇐⇒
+p + iTv − 2iTv′ = S(p − iTv)
+⇐⇒
+2iTv′ = 0
+=⇒
+v′ = 0.
+□
+The scattering operator has a subdomain-wise block diagonal structure. This is clearly visible
+from the formula S = Id + 2iTB(A − iB∗TB)−1B∗ where each term in the right hand side is
+block diagonal. This yields
+S = diag(SΓ, SΩ1, . . . , SΩJ)
+where SΩj = Id + 2iTΩjBΩj(AΩj − iB∗
+ΩjTΩjBΩj)−1B∗
+Ωj
+where SΓ = Id + 2iTΓBΓ(AΓ − iB∗
+ΓTΓBΓ)−1B∗
+Γ
+Let us discuss the particular form that takes the boundary scattering operator SΓ for Dirichlet,
+Neumann and Robin conditions. Recall that BΓ : Hb(Γ) := H1/2(Γ) × H−1/2(Γ) → H1/2(Γ) is
+defined by BΓ(α, p) = α hence B∗
+Γ(p) = (p, 0).
+Example 7.4 (Dirichlet condition). Taking the same notations as in Example 3.1 and 5.2,
+since B∗
+Γp = (p, 0) for all p ∈ H−1/2(Γ), we conclude that BΓ(AΓ − iB∗
+ΓTΓBΓ)−1B∗
+Γ = 0 and
+finally
+SΓ = +Id.
+Example 7.5 (Neumann condition). Taking the same notations as in Example 3.2 and
+5.3, in this situation we have BΓ(AΓ − iB∗
+ΓTΓBΓ)−1B∗
+Γ = iT−1
+Γ . This yields the expression
+SΓ = −Id.
+Example 7.6 (Robin condition). Taking the same notations as in Example 3.3 and 5.4, in
+this situation we have BΓ(AΓ − iB∗
+ΓTΓBΓ)−1B∗
+Γ = i(Λ + TΓ)−1 which yields
+SΓ = (Λ − TΓ)(Λ + TΓ)−1.
+8
+Skeleton formulation
+Now we shall use the scattering operator of the previous section to transform further the
+boundary value problem (35). Once volume unknowns have been eliminated, this reduces to
+an equation involving only traces on the skeleton of the subdomain partition.
+17
+
+Proposition 8.1.
+Assume (A1)-(A2)-(A3)-(A4). Define f ∈ H(Σ)∗ by f = −2iΠTB(A−iB∗TB)−1ℓ. If (u, p) ∈
+H(Ω) × H(Σ)∗ solves (35), then q = p − iTB(u) satisfies the skeleton problem
+q ∈ H(Σ)∗ and
+(Id + ΠS)q = f.
+(37)
+Reciprocally if q satisfies the above equation then the pair (u, p) ∈ H(Ω) × H(Σ)∗, given by
+u = (A − iB∗TB)−1(B∗q + ℓ) and p = q + iTB(u), solves (35).
+Proof:
+If (u, p) ∈ H(Ω) × H(Σ)∗ solves (35) and q = p − iTB(u), then (A − iB∗TB)u = B∗(p −
+iTBu) + ℓ. Left multiplying this equality by 2iTB(A − iB∗TB)−1 yields an expression for
+2iTB(u) that can be used in p+iTB(u) = q+2iTB(u) in the last line of (35). This eventually
+leads to (37).
+Reciprocally if q solves (37) and u = (A − iB∗TB)−1(B∗q + ℓ) and p = q + iTB(u), then we
+have Au = B∗(q + iTBu) + ℓ = B∗p + ℓ. On the other hand, using the expression of f and
+S, the skeleton equation in (37) writes
+q + Π(q + 2iTB(A − iB∗TB)−1(B∗q + ℓ)) = 0
+⇐⇒
+q + Π(q + 2iTB(u)) = 0
+⇐⇒
+p − iTB(u) + Π(p + iTB(u)) = 0
+This finally proves that the pair (u, p) satisfies (35)
+□
+Next we investigate whether or not the skeleton formulation (8.1) is uniquely solvable. We
+will show that this is directly correlated to the unique solvability of (14).
+Proposition 8.2.
+Assume (A1)-(A2)-(A3)-(A4).
+The application (v, p) �→ p − iT(v) induces a continuous
+isomorphism from C (A) ∩ X (Σ) onto ker(Id + ΠS). As a consequence
+dim( ker(Id + ΠS) ) = dim( ker(AΩ×Γ) ).
+Proof:
+First of all, if (v, p) ∈ C (A) ∩ X (Σ), then p + iTv = S(p − iTv) according to Lemma
+7.3, and p − iTv = −Π(p + iTv) according to (27). Combining these two identities leads to
+p − iTv ∈ ker(Id + ΠS). Next if (v, p) ∈ C (A) ∩ X (Σ) and p − iTv = 0, then (v, p) = (0, 0)
+according to Lemma 6.1 hence the injectivity.
+Finally if q ∈ ker(Id + ΠS), then there exists (v, p) ∈ C (A) unique such that p − iTv = q
+according to Lemma 6.1, and applying (36), we obtain S(q) = S(p−iTv) = p+iTv. From this
+later identity and (Id+ΠS)q = 0 leads to −p+iTv = Π(p+iTv) which implies (v, p) ∈ X (Σ)
+according to Proposition 4.3. Hence we conclude (v, p) ∈ C (A) ∩ X (Σ).
+□
+Proposition 8.3.
+Assume (A1)-(A2)-(A3)-(A4). The subspace range(Id + ΠS) is closed in H(Σ)∗.
+18
+
+Proof:
+Define Θ : H(Σ)∗ → H (Σ) by Θ(q) := (iT−1(q), q), which satisfies 2∥q∥2
+T−1 = ∥Θ(q)∥2
+T×T−1
+for all q ∈ H(Σ)∗. Taking account that C (A) + X (Σ) is closed, see Proposition 6.8, we are
+going to prove that
+range(Id + ΠS) = Θ−1(C (A) + X (Σ)).
+Take any p ∈ range(Id + ΠS). Applying Lemma 6.1, there exists a unique (v, q) ∈ C (A) such
+that 2p = (Id + ΠS)(q − iTv). Since S(q − iTv) = q + iTv according to Proposition 7.2, and
+writing 2p = (Id + Π)p + (Id − Π)p, we obtain
+(Id + Π)p + (Id − Π)p = q − iTv + Π(q + iTv)
+⇐⇒
+(Id + Π)p + (Id − Π)p = (Id + Π)q − (Id − Π)(iTv)
+⇐⇒
+(Id + Π)(p − q) = −(Id − Π)(p + iTv).
+As (Id ± Π)/2 are two mutually orthogonal projectors, see Proposition 4.3, we deduce on
+the one hand that (Id + Π)(p − q) = 0 and (Id − Π)(p + iTv) = 0. This eventually leads
+to p − q ∈ X(Σ)◦ and p + iTv ∈ T(X(Σ))
+⇐⇒
+iT−1p − v ∈ X(Σ). We conclude that
+Θ(p) − (v, q) ∈ X (Σ). Hence Θ(p) ∈ C (A) + X (Σ).
+Reciprocally pick an arbitrary p ∈ Θ−1(C (A)+X (Σ)). This means that Θ(p)−(v, q) ∈ X (Σ)
+for some (v, q) ∈ C (A). As a consequence (Id − Π)(p + iTv) = 0 and (Id + Π)(p − q) = 0.
+Adding these two equations, and taking account that q +iTv = S(q −iTv) according to (36),
+leads to
+(Id + Π)(p − q) = −(Id − Π)(p + iTv)
+⇐⇒
+(Id + Π)p + (Id − Π)p = q − iTv + Π(q + iTv)
+⇐⇒
+p = (Id + ΠS)(q − iTv).
+□
+Proposition 8.4.
+Assume (A1)-(A2)-(A3)-(A4). Then
+codim( range(Id + ΠS) ) = codim( range(AΩ×Γ) ).
+Proof:
+Since range(Id+ΠS) is closed according to Proposition 8.3, we deduce that codim( range(Id+
+ΠS) ) = dim( ker((Id + ΠS)∗) ). Proposition 4.3, in particular the characterization of Q =
+(Id + Π)/2 as a T−1-orthogonal projection, show that Π2 = Id and Π∗ = T−1ΠT, so we have
+(Id + ΠS)∗ = (TΠ∗)−1(Id + ΠTS∗T−1)TΠ∗.
+Setting ˜S := TS∗T−1, and noting that TΠ∗ : H(Σ) → H(Σ)∗ is an isomorphism, we have
+dim( ker((Id + ΠS)∗) ) = dim( ker(Id + Π˜S) ). Let us have a close look at ˜S, taking account of
+the formulas given by Proposition 7.2. Since T∗ = T, we obtain
+˜S = Id + 2iTB(A∗ − iB∗TB)−1B∗.
+We see that ˜S differs from S only in that A is replaced by A∗. As a consequence, we can apply
+Proposition 8.2, replacing AΩ×Γ with A∗
+Ω×Γ. Using (16), this yields dim( ker(Id + Π˜S) ) =
+dim( ker(A∗
+Ω×Γ) ) = codim( range(AΩ×Γ) ).
+□
+19
+
+If V1, V2 are Banach spaces, a bounded linear map L : V1 → V2 is of Fredholm type if and
+only if range(L) is closed in V2, dim( ker(L) ) < ∞ and codim( range(L) ) < ∞. In this case
+the index of L is the number index(L) := dim( ker(L) ) − codim( range(L) ). The results of the
+present paragraph (in particular Proposition 8.2, 8.3 and 8.4) lead to the following corollary.
+Corollary 8.5.
+Assume (A1)-(A2)-(A3)-(A4). The operator AΩ×Γ : H(Ω × Γ) → H(Ω × Γ)∗ is of Fredholm
+type if and only if Id + ΠS : H(Σ)∗ → H(Σ)∗ is of Fredholm type and, in this case, both
+operators have the same index.
+9
+Coercivity estimate
+Now we study quantitatively how the inf-sup constant of Id+ΠS relates to the inf-sup constant
+of the operator AΩ×Γ. Taking the cue from [6, §8], we first establish an intermediate result.
+Recall that inf-sup constants are defined according to (4).
+Proposition 9.1.
+Assume (A1)-(A2)-(A3)-(A4). Then
+infsup
+H(Ω×Γ)→H(Ω×Γ)∗(AΩ×Γ) ≤ (1 + ∥A∥)
+inf
+u∈C (A)\{0}
+v∈X (Σ)\{0}
+∥u + v∥T×T−1
+∥u∥T×T−1
+where
+∥A∥ :=
+sup
+u,v∈H(Ω)\{0}
+|⟨u, A(v)⟩|
+∥u∥H(Ω)∥v∥H(Ω)
+.
+Proof:
+In the case where C (A)∩X (Σ) ̸= {0}, the inf-sup constant vanishes since ker(AΩ×Γ) ̸= {0}
+according to Proposition 6.6. So the estimate is automatically satisfied in this case. We shall
+assume C (A) ∩ X (Σ) = {0}. According to Proposition 6.6 this leads to
+ker(AΩ×Γ) ̸= {0}
+α :=
+infsup
+H(Ω×Γ)→H(Ω×Γ)∗(AΩ×Γ) > 0.
+(38)
+Now pick any u ∈ C (A) \ {0} and any v ∈ X (Σ) \ {0}, and set (pd, pn) := u + v ∈ H (Σ) =
+H(Σ)×H(Σ)∗. The invertibility of AΩ×Γ provides the existence of a unique ϕ ∈ X(Ω) satisfying
+⟨A(ϕ), w⟩ = −⟨AB†(pd), w⟩ + ⟨pn, B(w)⟩ for all w ∈ X(Ω). In particular
+α ∥ϕ∥H(Ω) ≤ ∥A∥ ∥pd∥T + ∥pn∥T−1.
+(39)
+Set φ = ϕ+B†(pd) and ud = B(φ) = B(ϕ)+pd. By construction, for any w ∈ H(Ω) satisfying
+B(w) = 0 we have ⟨A(φ), w⟩ = ⟨pn, B(w)⟩ = 0, which rewrites A(φ) ∈ ker(B)◦. Applying
+i) of Lemma 4.1 we have Aφ = B∗un for some un ∈ H(Σ)∗.
+This implies in particular
+un = (BB†)∗un = (B†)∗B∗un = (B†)∗Aφ. From the previous definitions, and the fact that
+∥B(w)∥T ≤ ∥w∥H(Ω) and ∥B†(q)∥H(Ω) = ∥q∥T, we obtain the estimates
+∥φ∥H(Ω) ≤ ∥ϕ∥H(Ω) + ∥pd∥T
+∥ud∥T ≤ ∥φ∥H(Ω)
+∥un∥T−1 ≤ ∥A∥ ∥φ∥H(Ω).
+(40)
+20
+
+We have Aφ = B∗un and Bφ = ud hence (ud, un) ∈ C (A) by construction. On the other hand
+we have pd − ud = Bϕ ∈ X(Σ) since ϕ ∈ X(Ω) and, for any w ∈ X(Σ) we have B†(w) ∈ X(Ω)
+hence ⟨pn − un, w⟩ = ⟨Aφ, B†w⟩ − ⟨Aφ, B†w⟩ = 0, which implies that pn − un ∈ X(Σ)◦.
+Finally we have shown that (ud, un) ∈ C (A) and (pd, pn) − (ud, un) ∈ X (Σ) and, since
+p = u + v ∈ C (A) ⊕ X (Σ), we conclude that u = (ud, un). There only remains to combine
+(39) and (40) to obtain the desired estimate.
+□
+Theorem 9.2.
+Assume (A1)-(A2)-(A3)-(A4). Then
+infsup
+H(Ω×Γ)→H(Ω×Γ)∗(AΩ×Γ) ≤ (1 + ∥A∥)
+infsup
+H(Σ)∗→H(Σ)∗(Id + ΠS).
+Proof:
+In the case where ker(AΩ×Γ) ̸= {0} we also have ker(Id + ΠS) ̸= {0} according to Propo-
+sition 8.2 and, in this situation, the desired estimate is satisfied, with both sides of the es-
+timate equal to 0. Hence we can assume that ker(AΩ×Γ) = {0} and in this situation both
+AΩ×Γ : H(Ω × Γ) → H(Ω × Γ)∗ and Id + ΠS : H(Σ) → H(Σ)∗ are are injective with closed
+range. Pick an arbitrary f ∈ H(Σ)∗. According to Lemma 6.1, there exists a unique pair
+u = (ud, un) ∈ C (A) such that f = un − iT(ud) and we have ∥f∥T−1 ≤
+√
+2∥u∥T×T−1 which
+re-writes as
+∥u∥T×T−1
+∥f∥T−1×T
+≥
+1
+√
+2
+.
+Next set g = (Id + ΠS)f and p = (pd, pn) = (T−1(g), −ig)/2.
+We have in particular
+∥g∥T−1 =
+√
+2∥p∥T×T−1. Since S(f) = S(un−iT(ud)) = un+iT(ud) according to Proposition
+7.2, we obtain
+un − iT(ud) + Π(un + iT(ud)) = f + ΠS(f)
+= g = (Id + Π)g/2 + (Id − Π)g/2
+= (Id + Π)pn − i(Id − Π)T(pd)
+= pn − iT(pd) + Π(pn + iT(pd))
+Re-arranging the terms in the equality above so as to move all contributions involving Π in
+the right hand side, we obtain −(pn − un) + iT(pd − ud) = Π((pn − un) + iT(pd − ud)).
+According to Proposition 4.3, this implies that (pd, pn) − (ud, un) ∈ X (Σ). Since we have
+(ud, un) ∈ C (A) by construction, we can apply Proposition 9.1 which yields
+∥(Id + ΠS)f∥T−1
+∥f∥T−1
+= ∥g∥T−1
+∥f∥T−1 ≥ ∥p∥T×T−1
+∥u∥T×T−1 ≥
+infsup
+H(Ω×Γ)→H(Ω×Γ)∗(AΩ×Γ)/(1 + ∥A∥).
+This establishes the desired estimate, since this holds for any f ∈ H(Σ)∗.
+□
+The estimate provided by Theorem 9.2 is remarkable in several respects. First of all it holds
+even if ker(AΩ×Γ) is non-trivial. Secondly it does not involve any hidden “C > 0” constant.
+In particular it does not involve any frequency dependency, although the infsup constant of
+AΩ×Γ a priori depends itself on the frequency. This means that, to estimate the frequency
+dependency of the infsup constant of Id + ΠS, it suffices to derive such an estimate for AΩ×Γ.
+A further striking feature is that the number of subdomains J does not come into play in this
+estimate.
+21
+
+As an interesting additional result in the perspective of an effective linear solve, the contrac-
+tivity of Π and S leads to the coercivity of the operator Id + ΠS. The next result can be
+combined with Theorem 9.2 to obtain an effective estimate of the coercivity constant.
+Corollary 9.3.
+Assume (A1)-(A2)-(A3)-(A4). Then Id + ΠS : H(Σ)∗ → H(Σ)∗ is coercive with respect to the
+scalar product induced by T−1 and we have
+inf
+q∈H(Σ)∗\{0}
+ℜe{⟨(Id + ΠS)q, T−1q⟩}
+∥q∥2
+T−1
+≥ 1
+2
+�
+infsup
+H(Σ)∗→H(Σ)∗(Id + ΠS)
+�2.
+Proof:
+For any q ∈ H(Σ)∗,
+∥q∥2
+T−1 ≥ ∥ΠS(q)∥2
+T−1 = ∥(Id + ΠS)q − q∥2
+T−1
+∥q∥2
+T−1 ≥ ∥ΠS(q)∥2
+T−1 = ∥(Id + ΠS)q∥2
+T−1 + ∥q∥2
+T−1 − 2ℜe{⟨(Id + ΠS)q, T−1q⟩}
+=⇒
+ℜe{⟨(Id + ΠS)q, T−1q⟩}/∥q∥2
+T−1 ≥
+�
+∥(Id + ΠS)q∥T−1/∥q∥T−1
+�2/2.
+□
+We conclude this article illustrating how the previous results lead to estimations of the coer-
+civity constant of the skeleton operator for a concrete case.
+Example 9.4.
+Consider the case Rd = R2 or R3. Assume that µ = 1, κ = k ∈ (0, +∞), and choose AΓ as in
+Example 3.3 with ⟨Λ(u), v⟩ = k
+�
+Γ uvdσ which models the Robin condition ∂nu − iku = 0 on
+Γ. So we. Assume in addition that Ω is a convex polyhedron. Then we have
+⟨AΩ×Γ(u, p), (v, q)⟩ =
+�
+Ω
+∇u∇v − k2uvdx − ik
+�
+Γ
+uvdσ +
+�
+Γ
+qTΓp dσ.
+Let us take γ = 1/k for the parameter involved in (8). From these choices, and proceeding
+like in [15, Lem.2.4] for dealing with boundary terms on Γ, we see that the continuity modulus
+∥A∥ (as defined in Proposition 9.1) can be bounded independently of k. On the other hand,
+we know from [18] that
+infsup
+H(Ω×Γ)→H(Ω×Γ)∗(AΩ×Γ) ≥
+O
+k→∞(1/k).
+We can now plug this estimate into Theorem 9.2, and we see that the inf-sup constant of
+Id + ΠS admits also a lower bound that behaves like O(1/k) for k → ∞. Finally combining
+with Corollary 9.3, we see that the coercivity constant of the skeleton formulation behaves like
+O(1/k2) i.e.
+inf
+q∈H(Σ)∗\{0} ℜe{⟨(Id + ΠS)q, T−1q⟩}/∥q∥2
+T−1 ≥
+O
+k→∞(1/k2).
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+

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